(2218,97,'2.2.1 Examples','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>EXAMPLE&nbsp;2.8</strong></span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The following data are the heights (in inches to the nearest half inch) of 100 male semiprofessional soccer players. The heights are&nbsp;<strong><span style=\"color:#424242\">continuous</span></strong>&nbsp;data, since height is measured.</span></p>\n\n<p style=\"text-align:justify\"><br />\n<span style=\"font-size:14px\">60; 60.5; 61; 61; 61.5<br />\n63.5; 63.5; 63.5<br />\n64; 64; 64; 64; 64; 64; 64; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5; 64.5<br />\n66; 66; 66; 66; 66; 66; 66; 66; 66; 66; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 66.5; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67; 67.5; 67.5; 67.5; 67.5; 67.5; 67.5; 67.5<br />\n68; 68; 69; 69; 69; 69; 69; 69; 69; 69; 69; 69; 69.5; 69.5; 69.5; 69.5; 69.5<br />\n70; 70; 70; 70; 70; 70; 70.5; 70.5; 70.5; 71; 71; 71<br />\n72; 72; 72; 72.5; 72.5; 73; 73.5<br />\n74</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The smallest data value is 60. Since the data with the most decimal places has one decimal (for instance, 61.5), we want our starting point to have two decimal places. Since the numbers 0.5, 0.05, 0.005, etc. are convenient numbers, use 0.05 and subtract it from 60, the smallest value, for the convenient starting point.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">60 &ndash; 0.05 = 59.95 which is more precise than, say, 61.5 by one decimal place. The starting point is, then, 59.95.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The largest value is 74, so 74 + 0.05 = 74.05 is the ending value.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Next, calculate the width of each bar or class interval. To calculate this width, subtract the starting point from the ending value and divide by the number of bars (you must choose the number of bars you desire). Suppose you choose eight bars.</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span class=\"math-tex\">\\(\\frac{74.05-59.95}{8}=1.76\\)</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>NOTE </strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">We will round up to two and make each bar or class interval two units wide. Rounding up to two is one way to prevent a value from falling on a boundary. Rounding to the next number is often necessary even if it goes against the standard rules of rounding. For this example, using 1.76 as the width would also work. A guideline that is followed by some for the width of a bar or class interval is to take the square root of the number of data values and then round to the nearest whole number, if necessary. For example, if there are 150 values of data, take the square root of 150 and round to 12 bars or intervals.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The boundaries are:</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">59.95</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">59.95 + 2 = 61.95</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">61.95 + 2 = 63.95</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">63.95 + 2 = 65.95</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">65.95 + 2 = 67.95</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">67.95 + 2 = 69.95</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">69.95 + 2 = 71.95</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">71.95 + 2 = 73.95</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">73.95 + 2 = 75.95</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The heights 60 through 61.5 inches are in the interval 59.95&ndash;61.95. The heights that are 63.5 are in the interval 61.95&ndash;63.95. The heights that are 64 through 64.5 are in the interval 63.95&ndash;65.95. The heights 66 through 67.5 are in the interval 65.95&ndash;67.95. The heights 68 through 69.5 are in the interval 67.95&ndash;69.95. The heights 70 through 71 are in the interval 69.95&ndash;71.95. The heights 72 through 73.5 are in the interval 71.95&ndash;73.95. The height 74 is in the interval 73.95&ndash;75.95.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The following histogram displays the heights on the&nbsp;<em>x</em>-axis and relative frequency on the&nbsp;<em>y</em>-axis.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238249671623824967.jpg\" style=\"height:566px; width:800px\" /></span></p>\n\n<p style=\"text-align:center\"><span style=\"font-size:14px\"><strong><span style=\"color:#424242\">Figure&nbsp;2.5</span></strong></span></p>\n\n<div>\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><strong>Interval Frequency Relative Frequency</strong></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\">59.95&ndash;61.95&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;5&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;5/100 = 0.05</span></p>\n\n<p><span style=\"font-size:14px\">61.95&ndash;63.95 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;3 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;3/100 = 0.03</span></p>\n\n<p><span style=\"font-size:14px\">63.95&ndash;65.95 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;15&nbsp; &nbsp; &nbsp; 15/100 = 0.15</span></p>\n\n<p><span style=\"font-size:14px\">65.95&ndash;67.95 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;40&nbsp; &nbsp; &nbsp; &nbsp;40/100 = 0.40</span></p>\n\n<p><span style=\"font-size:14px\">67.95&ndash;69.95&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;17&nbsp; &nbsp; &nbsp; &nbsp;17/100 = 0.17</span></p>\n\n<p><span style=\"font-size:14px\">69.95&ndash;71.95 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;12 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;12/100 = 0.12</span></p>\n\n<p><span style=\"font-size:14px\">71.95&ndash;73.95&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;7&nbsp; &nbsp; &nbsp; 7/100 = 0.077</span></p>\n\n<p><span style=\"font-size:14px\">3.95&ndash;75.95&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; &nbsp;1&nbsp; &nbsp; &nbsp; &nbsp;&nbsp;1/100 = 0.01</span></p>\n</div>\n\n<p style=\"text-align:center\">&nbsp;</p>\n\n<div>\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>EXAMPLE&nbsp;2.9</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Create a histogram for the following data: the number of books bought by 50 part-time college students at ABC College. The number of books is&nbsp;<strong><span style=\"color:#424242\">discrete data</span></strong>, since books are counted.</span></p>\n\n<p style=\"text-align:justify\"><br />\n<span style=\"font-size:14px\">1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1<br />\n2; 2; 2; 2; 2; 2; 2; 2; 2; 2<br />\n3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3; 3<br />\n4; 4; 4; 4; 4; 4<br />\n5; 5; 5; 5; 5<br />\n6; 6</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Eleven students buy one book. Ten students buy two books. Sixteen students buy three books. Six students buy four books. Five students buy five books. Two students buy six books.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Calculate the width of each bar/bin size/interval size.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>Solution 2.10</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The smallest data value is 1, and the largest data value is 6. To make sure each is included in an interval, we can use 0.5 as the smallest value and 6.5 as the largest value by subtracting and adding 0.5 to these values. We have a small range here of 6 (6.5 &ndash;&ndash; 0.5), so we will want a fewer number of bins; let&rsquo;&#39;s say six this time. So, six divided by six bins gives a bin size (or interval size) of one.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Notice that we may choose different rational numbers to add to, or subtract from, our maximum and minimum values when calculating bin size. In the previous example, we added and subtracted .05, while this time, we added and subtracted .5. </span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Given a data set, you will be able to determine what is appropriate and reasonable.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The following histogram displays the number of books on the&nbsp;<em><span style=\"color:#424242\">x</span></em>-axis and the frequency on the&nbsp;<em><span style=\"color:#424242\">y</span></em>-axis.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Calculate the number of bars as follows:</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span class=\"math-tex\">\\(\\frac{6.5-0.5}{Number of bars}=1\\)</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">where 1 is the width of a bar. Therefore, bars = 6.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:#f9f9f9\"><span style=\"color:black\">The following histogram displays the number of books on the&nbsp;<em>x</em>-axis and the frequency on the&nbsp;<em>y</em>-axis.</span></span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"background-color:#f9f9f9\"><span style=\"color:black\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238301781623830178.jpg\" style=\"height:503px; width:800px\" /></span></span></span></p>\n\n<p style=\"text-align:center\"><span style=\"font-size:14px\"><strong><span style=\"color:#424242\">Figure&nbsp;2.6</span></strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>EXAMPLE&nbsp;2.10</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Using this data set, construct a histogram.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<table cellspacing=\"0\" class=\"Table\" style=\"border-collapse:collapse; width:900px\">\n	<caption>\n	<p><span style=\"font-size:20px\"><strong>Number of hours my classmates spent playing video games on weekends</strong></span></p>\n	</caption>\n	<tbody>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">9.95</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">10</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">2.25</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">16.75</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">0</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">19.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">22.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">7.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">15</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">12.75</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">5.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">11</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">10</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">20.75</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">17.5</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">23</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">21.9</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">24</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">23.75</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">18</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">20</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">15</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">22.9</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">18.8</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">20.5</span></div>\n			</td>\n		</tr>\n	</tbody>\n</table>\n\n<p style=\"text-align:center\"><span style=\"font-size:14px\"><strong><span style=\"color:#424242\">Table2.13</span></strong></span></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong><span style=\"color:black\">Solution&nbsp;1</span></strong></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><strong><span style=\"color:black\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238304301623830430.jpg\" style=\"height:686px; width:800px\" /></span></strong></span></p>\n\n<p style=\"text-align:center\"><span style=\"font-size:14px\"><strong><span style=\"color:#424242\">Figure&nbsp;2.7</span></strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Some values in this data set fall on boundaries for the class intervals. A value is counted in a class interval if it falls on the left boundary, but not if it falls on the right boundary. Different researchers may set up histograms for the same data in different ways. There is more than one correct way to set up a histogram.</span></p>\n</div>','','1','2.2.1',NULL,'1','2','1','0','1','2021-06-21 11:15:37','2021-06-21 16:45:37'),(2219,97,'2.2.2 Frequency Polygons','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Frequency polygons are analogous to line graphs, and just as line graphs make continuous data visually easy to interpret, so too do frequency polygons.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">To construct a frequency polygon, first examine the data and decide on the number of intervals, or class intervals, to use on the&nbsp;<em>x</em>-axis and&nbsp;<em>y</em>-axis. After choosing the appropriate ranges, begin plotting the data points. After all the points are plotted, draw line segments to connect them.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>EXAMPLE&nbsp;2.11</strong></span></span></p>\n\n<table cellspacing=\"0\" class=\"Table\" style=\"border-collapse:collapse; border:none; width:800px\">\n	<caption>\n	<p><span style=\"font-size:20px\"><strong>A frequency polygon was constructed from the frequency table below.</strong></span></p>\n	</caption>\n	<thead>\n		<tr>\n			<td colspan=\"4\" style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; height:16px; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Frequency distribution for calculus final test scores</strong></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:25px; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Lower bound</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:25px; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Upper bound</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:25px; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Frequency</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:25px; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Cumulative frequency</strong></span></div>\n			</td>\n		</tr>\n	</thead>\n	<tbody>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:26px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">49.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:26px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">59.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:26px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:26px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">5</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:25px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">59.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:25px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">69.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:25px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">10</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:25px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">15</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:26px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">69.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:26px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">79.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:26px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">30</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:26px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">45</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:25px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">79.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:25px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">89.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:25px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">40</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:25px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">85</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:26px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">89.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:26px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">99.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:26px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">15</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:26px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">100</span></div>\n			</td>\n		</tr>\n	</tbody>\n</table>\n\n<p style=\"text-align:center\"><span style=\"font-size:14px\"><strong>Table2.14</strong></span></p>\n\n<p style=\"text-align:center\">&nbsp;</p>\n\n<p><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238308331623830833.jpg\" style=\"height:607px; width:800px\" /></p>\n\n<p style=\"text-align:center\"><span style=\"font-size:11pt\"><strong><span style=\"color:#424242\">Figure&nbsp;</span></strong><strong><span style=\"color:#424242\">2.8</span></strong></span></p>\n\n<p style=\"text-align:center\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The first label on the&nbsp;<em><span style=\"color:#424242\">x</span></em>-axis is 44.5. This represents an interval extending from 39.5 to 49.5. Since the lowest test score is 54.5, this interval is used only to allow the graph to touch the&nbsp;<em><span style=\"color:#424242\">x</span></em>-axis. The point labeled 54.5 represents the next interval, or the first &ldquo;real&rdquo; interval from the table, and contains five scores. This reasoning is followed for each of the remaining intervals with the point 104.5 representing the interval from 99.5 to 109.5. Again, this interval contains no data and is only used so that the graph will touch the&nbsp;<em><span style=\"color:#424242\">x</span></em>-axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Frequency polygons are useful for comparing distributions. This is achieved by overlaying the frequency polygons drawn for different data sets.</span></span></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>EXAMPLE&nbsp;2.12</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">We will construct an overlay frequency polygon comparing the scores from&nbsp;</span><a href=\"https://openstax.org/books/introductory-business-statistics/pages/2-1-display-data#eip-id1166383300854\" style=\"color:blue; text-decoration:underline\"><span style=\"color:black\">Example 2.11</span></a><span style=\"color:black\">&nbsp;with the students&rsquo; final numeric grade.</span></span></p>\n\n<p>&nbsp;</p>\n\n<p>&nbsp;</p>\n\n<table cellspacing=\"0\" class=\"Table\" style=\"border-collapse:collapse; border:none; width:900px\">\n	<thead>\n		<tr>\n			<td colspan=\"4\" style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Frequency distribution for calculus final test scores</strong></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Lower bound</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Upper bound</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Frequency</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Cumulative frequency</strong></span></div>\n			</td>\n		</tr>\n	</thead>\n	<tbody>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">49.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">59.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">5</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">59.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">69.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">10</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">15</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">69.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">79.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">30</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">45</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">79.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">89.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">40</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">85</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">89.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">99.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">15</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">100</span></div>\n			</td>\n		</tr>\n	</tbody>\n</table>\n\n<p style=\"text-align:center\"><span style=\"font-size:20px\"><strong>Table2.16</strong></span></p>\n\n<table cellspacing=\"0\" class=\"Table\" style=\"border-collapse:collapse; border:none; width:900px\">\n	<thead>\n		<tr>\n			<td colspan=\"4\" style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Frequency distribution for calculus final grades</strong></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Lower bound</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Upper bound</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Frequency</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Cumulative frequency</strong></span></div>\n			</td>\n		</tr>\n	</thead>\n	<tbody>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">49.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">59.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">10</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">10</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">59.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">69.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">10</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">20</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">69.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">79.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">30</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">50</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">79.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">89.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">45</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">95</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">89.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">99.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">100</span></div>\n			</td>\n		</tr>\n	</tbody>\n</table>\n\n<p style=\"text-align:center\"><span style=\"font-size:14px\"><strong>Table</strong><strong>2.17</strong></span></p>\n\n<p><span style=\"font-size:14px\"><strong><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238310731623831073.jpg\" style=\"height:668px; width:800px\" /></strong></span></p>\n\n<p style=\"text-align:center\"><span style=\"font-size:11pt\"><strong><span style=\"color:#424242\">Figure&nbsp;</span></strong><strong><span style=\"color:#424242\">2.9</span></strong> </span></p>','','1','2.2.2',NULL,'1','2','1','0','1','2021-06-21 11:19:09','2021-06-21 16:49:09'),(2220,97,'2.2.3 Constructing a Time Series Graph','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Suppose that we want to study the temperature range of a region for an entire month. Every day at noon we note the temperature and write this down in a log. A variety of statistical studies could be done with these data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">One feature of the data that we may want to consider is that of time. Since each date is paired with the temperature reading for the day, we don&lsquo;t have to think of the data as being random. We can instead use the times given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing temperature as the month progresses is called a time series graph.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p><span style=\"font-size:14px\">Constructing a Time Series Graph</span></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">To construct a time series graph, we must look at both pieces of our&nbsp;<strong>paired data set</strong>. We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By doing this, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur.</span></span></p>','','1','2.2.3',NULL,'1','3','1','0','1','2021-06-21 11:19:51','2021-06-21 16:49:51'),(2221,97,'2.2.4 Examples','','single','html','<p style=\"text-align:justify\"><span style=\"color:#000000\"><span style=\"font-size:14px\"><strong>EXAMPLE&nbsp;2.13</strong></span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The following data shows the Annual Consumer Price Index, each month, for ten years. Construct a time series graph for the Annual Consumer Price Index data only.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<table cellspacing=\"0\" class=\"Table\" style=\"border-collapse:collapse; border:none; width:900px\">\n	<caption>\n	<p style=\"text-align:justify\"><span style=\"font-size:18px\"><strong>Table2.18</strong></span></p>\n	</caption>\n	<thead>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom; width:78px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Year</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom; width:64px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Jan</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Feb</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Mar</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Apr</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>May</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Jun</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Jul</strong></span></div>\n			</td>\n		</tr>\n	</thead>\n	<tbody>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; width:78px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>2003</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:64px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">181.7</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">183.1</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">184.2</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">183.8</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">183.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">183.7</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">183.9</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; width:78px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>2004</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:64px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">185.2</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">186.2</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">187.4</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">188.0</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">189.1</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">189.7</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">189.4</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; width:78px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>2005</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:64px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">190.7</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">191.8</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">193.3</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">194.6</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">194.4</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">194.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">195.4</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; width:78px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>2006</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:64px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">198.3</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">198.7</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">199.8</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">201.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">202.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">202.9</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">203.5</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; width:78px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>2007</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:64px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">202.416</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">203.499</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">205.352</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">206.686</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">207.949</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">208.352</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">208.299</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; width:78px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>2008</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:64px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">211.080</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">211.693</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">213.528</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">214.823</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">216.632</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">218.815</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">219.964</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; width:78px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>2009</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:64px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">211.143</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">212.193</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">212.709</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">213.240</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">213.856</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">215.693</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">215.351</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; width:78px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>2010</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:64px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">216.687</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">216.741</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">217.631</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">218.009</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">218.178</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">217.965</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">218.011</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; width:78px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>2011</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:64px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">220.223</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">221.309</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">223.467</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">224.906</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">225.964</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">225.722</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">225.922</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; width:78px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>2012</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:64px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">226.665</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">227.663</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">229.392</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">230.085</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">229.815</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">229.478</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">229.104</span></div>\n			</td>\n		</tr>\n	</tbody>\n</table>\n\n<p>&nbsp;</p>\n\n<table cellspacing=\"0\" class=\"Table\" style=\"border-collapse:collapse; border:none; width:900px\">\n	<caption>\n	<p><span style=\"font-size:18px\"><strong>Table2.19</strong></span></p>\n	</caption>\n	<thead>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Year</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Aug</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Sep</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Oct</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Nov</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Dec</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Annual</strong></span></div>\n			</td>\n		</tr>\n	</thead>\n	<tbody>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>2003</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">184.6</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">185.2</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">185.0</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">184.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">184.3</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">184.0</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>2004</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">189.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">189.9</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">190.9</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">191.0</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">190.3</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">188.9</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>2005</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">196.4</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">198.8</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">199.2</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">197.6</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">196.8</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">195.3</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>2006</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">203.9</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">202.9</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">201.8</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">201.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">201.8</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">201.6</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>2007</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">207.917</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">208.490</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">208.936</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">210.177</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">210.036</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">207.342</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>2008</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">219.086</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">218.783</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">216.573</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">212.425</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">210.228</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">215.303</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>2009</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">215.834</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">215.969</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">216.177</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">216.330</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">215.949</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">214.537</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>2010</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">218.312</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">218.439</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">218.711</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">218.803</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">219.179</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">218.056</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>2011</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">226.545</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">226.889</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">226.421</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">226.230</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">225.672</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">224.939</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>2012</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">230.379</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">231.407</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">231.317</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">230.221</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">229.601</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">229.594</span></div>\n			</td>\n		</tr>\n	</tbody>\n</table>\n\n<p>&nbsp;</p>\n\n<p>&nbsp;</p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238315721623831572.jpg\" style=\"height:312px; width:800px\" /></span></p>\n\n<p style=\"text-align:center\"><strong><span style=\"font-size:14px\">Fig 2.10</span></strong></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>Uses of a Time Series Graph</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Time series graphs are important tools in various applications of statistics. When recording values of the same variable over an extended period of time, sometimes it is difficult to discern any trend or pattern. However, once the same data points are displayed graphically, some features jump out. Time series graphs make trends easy to spot</span>.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>How NOT to Lie with Statistics</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">It is important to remember that the very reason we develop a variety of methods to present data is to develop insights into the subject of what the observations represent. We want to get a &quot;sense&quot; of the data. Are the observations all very much alike or are they spread across a wide range of values, are they bunched at one end of the spectrum or are they distributed evenly and so on. We are trying to get a visual picture of the numerical data. Shortly we will develop formal mathematical measures of the data, but our visual graphical presentation can say much. It can, unfortunately, also say much that is distracting, confusing and simply wrong in terms of the impression the visual leaves. Many years ago Darrell Huff wrote the book&nbsp;<em>How to Lie with Statistics</em>. It has been through 25 plus printings and sold more than one and one-half million copies. His perspective was a harsh one and used many actual examples that were designed to mislead. He wanted to make people aware of such deception, but perhaps more importantly to educate so that others do not make the same errors inadvertently.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Again, the goal is to enlighten with visuals that tell the story of the data. Pie charts have a number of common problems when used to convey the message of the data. Too many pieces of the pie overwhelm the reader. More than perhaps five or six categories ought to give an idea of the relative importance of each piece. This is after all the goal of a pie chart, what subset matters most relative to the others. If there are more components than this then perhaps an alternative approach would be better or perhaps some can be consolidated into an &quot;other&quot; category. Pie charts cannot show changes over time, although we see this attempted all too often. In federal, state, and city finance documents pie charts are often presented to show the components of revenue available to the governing body for appropriation: income tax, sales tax motor vehicle taxes and so on. In and of itself this is interesting information and can be nicely done with a pie chart. The error occurs when two years are set side-by-side. Because the total revenues change year to year, but the size of the pie is fixed, no real information is provided and the relative size of each piece of the pie cannot be meaningfully compared.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Histograms can be very helpful in understanding the data. Properly presented, they can be a quick visual way to present probabilities of different categories by the simple visual of comparing relative areas in each category. Here the error, purposeful or not, is to vary the width of the categories. This of course makes comparison to the other categories impossible. It does embellish the importance of the category with the expanded width because it has a greater area, inappropriately, and thus visually &quot;says&quot; that that category has a higher probability of occurrence.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Time series graphs perhaps are the most abused. A plot of some variable across time should never be presented on axes that change part way across the page either in the vertical or horizontal dimension. Perhaps the time frame is changed from years to months. Perhaps this is to save space or because monthly data was not available for early years. In either case this confounds the presentation and destroys any value of the graph. If this is not done to purposefully confuse the reader, then it certainly is either lazy or sloppy work.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Changing the units of measurement of the axis can smooth out a drop or accentuate one. If you want to show large changes, then measure the variable in small units, penny rather than thousands of dollars. And of course to continue the fraud, be sure that the axis does not begin at zero, zero. If it begins at zero, zero, then it becomes apparent that the axis has been manipulated.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Perhaps you have a client that is concerned with the volatility of the portfolio you manage. An easy way to present the data is to use long time periods on the time series graph. Use months or better, quarters rather than daily or weekly data. If that doesn&#39;t get the volatility down then spread the time axis relative to the rate of return or portfolio valuation axis. If you want to show &quot;quick&quot; dramatic growth, then shrink the time axis. Any positive growth will show visually &quot;high&quot; growth rates. Do note that if the growth is negative then this trick will show the portfolio is collapsing at a dramatic rate.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Again, the goal of descriptive statistics is to convey meaningful visuals that tell the story of the data. Purposeful manipulation is fraud and unethical at the worst, but even at its best, making these type of errors will lead to confusion on the part of the analysis.</span></span></p>','','1','2.2.4',NULL,'1','2','1','0','1','2021-06-21 11:21:58','2021-06-21 16:51:58'),(2222,97,'2.3 Measures of the Location of the Data','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The common measures of location are </span><strong><span style=\"color:black\">quartiles </span></strong><span style=\"color:black\">and </span><strong><span style=\"color:black\">percentiles</span></strong><span style=\"color:black\">.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Quartiles are special percentiles.</span></span></p>\n\n<ul>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">The </span><strong><span style=\"color:black\">first quartile</span></strong><span style=\"color:black\">, </span><em><span style=\"color:black\">Q</span></em><span style=\"color:black\">1, is the same as the 25th percentile, </span></span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:#373d3f\">The <strong>second quartile</strong>,&nbsp;<em>Q</em></span></span><em><sub><span style=\"color:#373d3f\">2</span></sub></em><span style=\"background-color:white\"><span style=\"color:#373d3f\">, is the&nbsp;same as the 50th percentile / median.</span></span></span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">The <strong>third quartile</strong>, </span><em><span style=\"color:black\">Q</span></em><span style=\"color:black\">3, is the same as the 75th percentile. </span></span></p>\n	</li>\n</ul>\n\n<p style=\"margin-left:52px; text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The median, </span><em><span style=\"color:black\">M</span></em><span style=\"color:black\">, is called both the second quartile and the 50th percentile.</span></span></p>\n\n<p style=\"margin-left:52px; text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">To calculate quartiles and percentiles, you must order the data from smallest to largest. Quartiles divide ordered data into quarters. Percentiles divide ordered data into hundredths. Recall that a percent means one-hundredth. So, percentiles mean the data is divided into 100 sections. To score in the 90th percentile of an exam does not mean, necessarily, that you received 90 percent on a test. It means that 90 percent of test scores are the same as or less than your score and that 10 percent of the </span>test scores are the same as or greater than your test score.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Percentiles are useful for comparing values. For this reason, universities and colleges use percentiles extensively. One instance in which colleges and universities use percentiles is when SAT results are used to determine a minimum testing score that will be used as an acceptance factor. For example, suppose Duke accepts SAT scores at or above the 75<sup>th</sup> percentile. That translates into a score of at least 1220.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Percentiles are mostly used with very large populations. Therefore, if you were to say that 90 percent of the test scores are less, and not the same or less, than your score, it would be acceptable because removing one particular data value is not significant.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The <strong>median </strong>is a number that measures the <em>center </em>of the data. You can think of the median as the <em>middle value</em>, bu t it does not actually have to be one of the observed values. It is a number that separates ordered data into halves. Half the values are the same number or smaller than the median, and half the values are the same number or larger. For example, consider the following data:</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">1, 11.5, 6, 7.2, 4, 8, 9, 10, 6.8, 8.3, 2, 2, 10, 1</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Ordered from smallest to largest:</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">When a data set has an even number of data values, the median is equal to the average of the two middle values when the data are arranged in ascending order (least to greatest). When a data set has an odd number of data values, the median is equal to the middle value when the data are arranged in ascending order.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Since there are 14 observations (an even number of data values), the median is between the seventh value, 6.8, and the eighth value, 7.2. To find the median, add the two values together and divide by two.</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span class=\"math-tex\">\\(\\frac{6.8+7.2}{2}=7\\)</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The median is seven. Half of the values are smaller than seven and half of the values are larger than seven.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Quartiles are numbers that separate the data into quarters. Quartiles may or may not be part of the data. To find the quartiles, first find the median, or second, quartile. The first quartile, Q1, is the middle value of the lower half of the data, and the third quartile, Q3, is<span style=\"color:black\"> the middle value, or median, of the upper half of the data. To get the idea, consider the same data set:</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The data set has an even number of values (14 data values), so the median will be the average of the two middle values (the average of 6.8 and 7.2), which is calculated as (6.8 + 7.2)/2 and equals 7.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">So, the median, or second quartile ( Q2 ), is 7.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The first quartile is the median of the lower half of the data, so if we divide the data into seven values in the lower half and seven values in the upper half, we can see that we have an odd number of values in the lower half. Thus, the median of the lower half, or the first quartile ( Q1 ) will be the middle value, or 2. Using the same procedure, we can see that the media&nbsp; of the upper half, or the third quartile ( Q3 ) will be the middle value of the upper half, or 9.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The quartiles are illustrated below:</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238318861623831886.jpg\" style=\"height:182px; width:800px\" /></span></p>\n\n<p style=\"text-align:center\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><strong><span style=\"color:#002060\">Figure&nbsp;2.11</span></strong></span></span></p>\n\n<p>&nbsp;</p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The <strong>interquartile range </strong>is a number that indicates the spread of the middle half, or the middle 50 percent of the data. It is the difference between the third quartile (<em>Q</em>3) and the first quartile (<em>Q</em>1)</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>IQR </em>= <em>Q</em>3 &ndash; <em>Q</em>1. The <em>IQR </em>for this data set is calculated as 9 minus 2, or 7.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The <em>IQR </em>can help to determine potential <strong>outliers</strong>. <strong>A value is suspected to be a potential outlier if it is less than 1.5 &times;</strong><strong><em>IQR </em></strong><strong>below the first quartile or more than 1.5 &times; </strong><strong><em>IQR </em></strong><strong>above the third quartile</strong>. Potential outliers always require further investigation.</span></p>','','1','2.3',NULL,'1','2','1','0','1','2021-06-21 11:24:01','2021-06-21 16:54:01'),(2223,97,'2.3.1 Examples','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Example 2.15</strong></span><span style=\"color:white\">15</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:white\">Example 2.15</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">For the following 13 real estate prices, calculate the </span><em><span style=\"color:black\">IQR </span></em><span style=\"color:black\">and determine if any prices are potential outliers. Prices are in dollars.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">389,950; 230,500; 158,000; 479,000; 639,000; 114,950; 5,500,000; 387,000; 659,000; 529,000; 575,000;</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">488,800; 1,095,000</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution 2.15</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Order the following data from smallest to largest:<br />\n114,950; 158,000; 230,500; 387,000; 389,950; 479,000; 488,800; 529,000; 575,000; 639,000; 659,000; 1,095,000; 5,500,000</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>M</em>&nbsp;= 488,800</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>Q</em><sub>1</sub>&nbsp;=&nbsp;(230,500&nbsp;+&nbsp;387,000)/2&nbsp; </span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&nbsp; &nbsp; &nbsp; = (230,500&nbsp;+&nbsp;387,000)/2&nbsp;</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&nbsp; &nbsp; &nbsp; = 308,750</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>Q</em><sub>3</sub>&nbsp;=&nbsp;(639,000&nbsp;+&nbsp;659,000)/2 </span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&nbsp; &nbsp; &nbsp; &nbsp;=(639,000&nbsp;+&nbsp;659,000)/2&nbsp;</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&nbsp; &nbsp; &nbsp; &nbsp;= 649,000</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>IQR</em>&nbsp;= 649,000 &ndash; 308,750 </span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&nbsp; &nbsp; &nbsp; &nbsp;= 340,250</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">(1.5)(<em>IQR</em>) = (1.5)(340,250)</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;= 510,375</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>Q</em><sub>1</sub>&nbsp;&ndash; (1.5)(<em>IQR</em>) = 308,750 &ndash; 510,375</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; = &ndash;201,625</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>Q</em><sub>3</sub>&nbsp;+ (1.5)(<em>IQR</em>) = 649,000 + 510,375</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; = 1,159,375</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">No house price is less than &ndash;201,625. However, 5,500,000 is more than 1,159,375. Therefore, 5,500,000 is a potential&nbsp;<strong>outlier</strong>.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Example 2.16</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Fifty statistics students were asked how much sleep they get per school night (rounded to the nearest hour). The results were as follows:</span></span></p>\n\n<table align=\"center\" cellspacing=\"0\" class=\"Table\" style=\"background:#ededed; border-collapse:collapse; border:none; width:900px\">\n	<caption>\n	<p><span style=\"font-size:18px\"><strong>Table2.24</strong></span></p>\n	</caption>\n	<thead>\n		<tr>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; height:40px; vertical-align:bottom; width:146px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">Amount of Sleep per School Night (Hours)</span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:40px; vertical-align:bottom; width:100px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">Frequency</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:40px; vertical-align:bottom; width:170px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">Relative Frequency</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:40px; vertical-align:bottom; width:185px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">Cumulative Relative Frequency</span></span></div>\n			</td>\n		</tr>\n	</thead>\n	<tbody>\n		<tr>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:10px; width:146px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">4</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:10px; width:100px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">2</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:10px; width:170px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.04</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:10px; width:185px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.04</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:10px; width:146px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">5</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:10px; width:100px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">5</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:10px; width:170px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">.10</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:10px; width:185px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.14</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:18px; width:146px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">6</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:18px; width:100px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">7</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:18px; width:170px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.14</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:18px; width:185px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">.28</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:10px; width:146px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">7</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:10px; width:100px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">12</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:10px; width:170px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.24</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:10px; width:185px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.52</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:19px; width:146px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">8</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:19px; width:100px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">14</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:19px; width:170px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.28</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:19px; width:185px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.80</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:9px; width:146px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">9</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:9px; width:100px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">7</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:9px; width:170px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.14</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:9px; width:185px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.94</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:10px; width:146px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">10</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:10px; width:100px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">3</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:10px; width:170px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.06</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:10px; width:185px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">1.00</span></span></div>\n			</td>\n		</tr>\n	</tbody>\n</table>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong><span style=\"color:black\">Find the 28th percentile</span></strong><span style=\"color:black\">. Notice the .28 in the Cumulative Relative Frequency column. Twenty-eight percent of 50 data values is 14 values. There are 14 values less than the 28th percentile. They include the two 4s, the five 5, and the seven 6s. The 28th percentile is between the last six and the first seven. </span><strong><span style=\"color:black\">The 28th percentile is 6.5.</span></strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong><span style=\"color:black\">Find the median</span></strong><span style=\"color:black\">. Look again at the Cumulative Relative Frequency column and find .52. The median is the 50<sup>th</sup> percentile or the second quartile. Fifty percent of 50 is 25. There are 25 values less than the median. They include the two 4s, the five 5s, the seven 6s, and 11 of the 7s. The median or 50th percentile is between the 25th, or seven, and 26th, or seven, values. </span><strong><span style=\"color:black\">The median is seven.</span></strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong><span style=\"color:black\">Find the third quartile</span></strong><span style=\"color:black\">. The third quartile is the same as the 75th percentile. You can </span><em><span style=\"color:black\">eyeball </span></em><span style=\"color:black\">this answer. If you look at the Cumulative Relative Frequency column, you find .52 and .80. When you have all the fours, fives, sixes, and sevens, you have 52 percent of the data. When you include all the 8s, you have 80 percent of the data.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong><span style=\"color:black\">The 75th percentile, then, must be an eight</span></strong><span style=\"color:black\">. Another way to look at the problem is to find 75 percent of 50, which is 37.5, and round up to 38. The third quartile, </span><em><span style=\"color:black\">Q</span></em><span style=\"color:black\">3, is the 38th value, which is an eight. You can check this answer by counting the values. There are 37 values below the third quartile and 12 values above.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Example 2.17</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Using </span><strong><span style=\"color:#26397f\">Table 2.24</span></strong><span style=\"color:black\">:</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">a. Find the 80th percentile.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">b. Find the 90th percentile.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">c. Find the first quartile. What is another name for the first quartile?</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution 2.17</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Using the data from the frequency table, we have the following:</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">a. The 80th percentile is between the last eight and the first nine in the table (between the 40th and 41st values).</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Therefore, we need to take the mean of the 40th an 41st values. The 80th percentile = (8 + 9)2 = 8.5.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">b. The 90th percentile will be the 45th data value (location is 0.90(50) = 45), and the 45th data value is nine.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">c. </span><em><span style=\"color:black\">Q</span></em><span style=\"color:black\">1 is also the 25th percentile. The 25th percentile location calculation: </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">25 = .25(50) = 12.5 &asymp; 13, the 13<sup>th</sup> data value. Thus, the 25th percentile is six.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>A Formula for Finding the <em>k</em>th Percentile</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">If you were to do a little research, you would find several formulas for calculating the </span><em><span style=\"color:black\">k</span></em><span style=\"color:black\">th percentile. Here is one of them.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">k </span></em><span style=\"color:black\">= the </span><em><span style=\"color:black\">k</span></em><span style=\"color:black\">th percentile. It may or may not be part of the data.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">i </span></em><span style=\"color:black\">= the index (ranking or position of a data value)</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">n </span></em><span style=\"color:black\">= the total number of data</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; Order the data from smallest to largest.</span></span></p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238322511623832251.png\" style=\"height:189px; width:800px\" /></span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; If </span><em><span style=\"color:black\">i </span></em><span style=\"color:black\">is an integer, then the </span><em><span style=\"color:black\">k</span></em><span style=\"color:black\">th percentile is the data value in the </span><em><span style=\"color:black\">i</span></em><span style=\"color:black\">th position in the ordered set of data.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; If </span><em><span style=\"color:black\">i </span></em><span style=\"color:black\">is not an integer, then round </span><em><span style=\"color:black\">i </span></em><span style=\"color:black\">up and round </span><em><span style=\"color:black\">i </span></em><span style=\"color:black\">down to the nearest integers. Average the two data values in these two </span>positions<span style=\"color:black\"> in the ordered data set. The formula and calculation are easier to understand in an example.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Example 2.18</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">i </span></em><span style=\"color:black\">= the index</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">n </span></em><span style=\"color:black\">= 29</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238323461623832346.png\" style=\"height:36px; width:238px\" /></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">This equation tells us that </span><em><span style=\"color:black\">i</span></em><span style=\"color:black\">, or the position of the data value in the data set, is 21. So, we will count over to the 21st position, which shows a data value of 64.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">b. </span><em><span style=\"color:black\">k </span></em><span style=\"color:black\">= 83rd percentile</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">i </span></em><span style=\"color:black\">= the index</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">n </span></em><span style=\"color:black\">= 29</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">i </span></em><span style=\"color:black\">= </span><em><span style=\"color:black\">k</span></em></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238324091623832409.png\" style=\"height:36px; width:238px\" /></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">which is </span><strong><span style=\"color:black\">not </span></strong><span style=\"color:black\">an integer. Round it down to 24 and up to 25. The age in the 24th position is 71, and the age in the 25th position is 72. Average 71 and 72. The 83rd percentile is 71.5 years.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Listed are 29 ages for Academy Award-winning best actors <em>in order from smallest to largest:</em></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">18, 21, 22, 25, 26, 27, 29, 30, 31, 33, 36, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">a. Find the 70th percentile.</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">b. Find the 83rd percentile.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution 2.18</strong></span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">a.</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>k</em>&nbsp;= 70</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>i</em>&nbsp;= the index</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>n</em>&nbsp;= 29</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span class=\"math-tex\">\\(i=(\\frac{k}{100}(n+1)=(\\frac{70}{100})(29+1)=21\\)</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">This equation tells us that&nbsp;<em>i</em>, or the position of the data value in the data set, is 21. So, we will count over to the 21<sup>st</sup>&nbsp;position, which shows a data value of 64.</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">b.</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>k</em>&nbsp;= 83<sup>rd</sup>&nbsp;percentile</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>i</em>&nbsp;= the index</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>n</em>&nbsp;= 29</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span class=\"math-tex\">\\(i=(\\frac{k}{100}(n+1)=(\\frac{83}{100})(29+1)=24.9\\)</span>&nbsp;, which is&nbsp;<strong>not</strong>&nbsp;an integer. Round it down to 24 and up to 25. The age in the 24<sup>th</sup>&nbsp;position is 71, and the age in the 25<sup>th</sup>&nbsp;position is 72. Average 71 and 72. The 83<sup>rd</sup>&nbsp;percentile is 71.5 years.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>A Formula for Finding the Percentile of a Value in a Data Set</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; Order the data from smallest to largest.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; </span><em><span style=\"color:black\">x </span></em><span style=\"color:black\">= the number of data values counting from the bottom of the data list up to but not including the data value for which</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">you want to find the percentile.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; </span><em><span style=\"color:black\">y </span></em><span style=\"color:black\">= the number of data values equal to the data value for which you want to find the percentile.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; </span><em><span style=\"color:black\">n </span></em><span style=\"color:black\">= the total number of data.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; Calculate&nbsp;&nbsp;</span><span class=\"math-tex\">\\(\\frac{x+5y}{n}(100)\\)</span>&nbsp;<span style=\"color:black\">Then round to the nearest integer.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>EXAMPLE&nbsp;2.19</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Listed are 29 ages for Academy Award-winning best actors&nbsp;<em>in order from smallest to largest:</em><br />\n18, 21, 22, 25, 26, 27, 29, 30, 31, 33, 36, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77</span></p>\n\n<ol style=\"list-style-type:lower-alpha\">\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\">Find the percentile for 58.</span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\">Find the percentile for 25.</span></p>\n	</li>\n</ol>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution&nbsp;1</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<ol style=\"list-style-type:lower-alpha\">\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\">Counting from the bottom of the list, there are 18 data values less than 58. There is one value of 58. </span></p>\n	</li>\n</ol>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;x</em>&nbsp;= 18 and&nbsp;<em>y</em>&nbsp;= 1.<span class=\"math-tex\">\\(\\frac{x+5y}{n}(100)=\\frac{18+5(1)}{29}(100)=63.80\\)</span>. Fifty-eight is the 64<sup>th</sup>&nbsp;percentile.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"margin-left:24px; text-align:justify\"><span style=\"font-size:14px\">b) Counting from the bottom of the list, there are three data values less than 25. There is one value of 25. </span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;x</em>&nbsp;= 3 and&nbsp;<em>y</em>&nbsp;= 1.&nbsp;<span class=\"math-tex\">\\(X+\\frac{x+5y}{n}(100)=\\frac{3+5(1)}{29}(100)=12.07\\)</span></span><span style=\"font-size:11pt\"><span style=\"font-size:12.0pt\">.<span style=\"font-size:14px\"> </span></span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; Twenty-five is the 12<sup>th</sup>&nbsp;percentile.</span></p>','','1','2.3.1',NULL,'1','2','1','0','1','2021-06-22 06:00:57','2021-06-22 11:30:57'),(2224,97,'2.3.2 Interpreting Percentiles, Quartiles, and Median','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\">A percentile indicates the relative standing of a data value when data are sorted into numerical order from smallest to largest.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Percentages of data values are less than or equal to the <em>p</em>th percentile. For example, 15 percent of data values are less than or equal to the 15th percentile.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"margin-left:48px; text-align:justify\"><span style=\"font-size:14px\">&bull; Low percentiles always correspond to lower data values.</span></p>\n\n<p style=\"margin-left:48px; text-align:justify\"><span style=\"font-size:14px\">&bull; High percentiles always correspond to higher data values.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">A percentile may or may not correspond to a value judgment about whether it is <em>good </em>or <em>bad</em>. The interpretation of whether a certain percentile is <em>good </em>or <em>bad </em>depends on the context of the situation to which the data apply. In some situations, a low percentile would be considered <em>good; </em>in other contexts, a high percentile might be considered <em>good</em>. In many situations, there is no value judgment that applies. A high percentile on a standardized test is considered good, while a lower percentile on body mass index might be considered good. A percentile associated with a person&#39;s height doesn&#39;t carry any value judgment.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Understanding how to interpret percentiles properly is important not only when describing data, but also when calculating probabilities in later chapters of this text.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\">&nbsp;</p>','','1','2.3.2',NULL,'1','2','1','0','1','2021-06-22 06:01:56','2021-06-22 11:31:56'),(2225,97,'2.3.3 Examples','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Example 2.20</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">On a timed math test, the first quartile for time it took to finish the exam was 35 minutes. Interpret the first quartile in the context of this situation.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution 2.20</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&bull; Twenty-five percent of students finished the exam in 35 minutes or less.</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&bull; Seventy-five percent of students finished the exam in 35 minutes or more.</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&bull; A low percentile could be considered good, as finishing more quickly on a timed exam is desirable. If you take too long, you might not be able to finish.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Example 2.21</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">On a 20-question math test, the 70th percentile for number of correct answers was 16. Interpret the 70th percentile in the context of this situation.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution 2.21</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&bull; Seventy percent of students answered 16 or fewer questions correctly.</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; Thirty percent of students answered 16 or more questions correctly.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; A higher percentile could be considered good, as answering more questions correctly is desirable.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Example 2.22</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">At a high school, it was found that the 30th percentile of number of hours that students spend studying per week is seven hours. Interpret the 30th percentile in the context of this situation.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution 2.22</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; Thirty percent of students study seven or fewer hours per week.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; Seventy percent of students study seven or more hours per week.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; In this example, there is not necessarily a </span><em><span style=\"color:black\">good </span></em><span style=\"color:black\">or </span><em><span style=\"color:black\">bad </span></em><span style=\"color:black\">value judgment associated with a higher or lower percentile, since the time a student studies per week is dependent on his/her needs</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Example 2.23</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">A middle school is applying for a grant that will be used to add fitness equipment to the gym. The principal surveyed 15 anonymous students to determine how many minutes a day the students spend exercising. The results from the 15 anonymous students are shown:</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">0 minutes, 40 minutes, 60 minutes, 30 minutes, 60 minutes,</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">10 minutes, 45 minutes, 30 minutes, 300 minutes, 90 minutes,</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">30 minutes, 120 minutes, 60 minutes, 0 minutes, 20 minutes</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Find the five values that make up the five number summary.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Min = 0</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Q1 = 20</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Med = 40</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Q3 = 60</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Max = 300</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Listing the data in ascending order gives the following:</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238333271623833327.png\" style=\"height:36px; width:238px\" /></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Figure 2.12</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The minimum value is 0.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The maximum value is 300.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Since there are an odd number of data values, the median is the middle value of this data set as it is arranged in ascending order, or 40.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The first quartile is the median of the lower half of the scores and does not include the median. The lower half has seven data values; the median of the lower half will equal the middle value of the lower half, or 20.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The third quartile is the median of the upper half of the scores and does not include the median. The upper half also has seven data values; so the median of the upper half will equal the middle value of the upper half, or 60.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">If you were the principal, would you be justified in purchasing new fitness equipment? Since 75 percent of the students exercise for 60 minutes or less daily, and since the </span><em><span style=\"color:black\">IQR </span></em><span style=\"color:black\">is 40 minutes (60 &ndash; 20 = 40), we know that half of the students surveyed exercise between 20 minutes and 60 minutes daily. This seems a reasonable amount of time spent exercising, so the principal would be justified in purchasing the new equipment.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">However, the principal needs to be careful. The value 300 appears to be a potential outlier.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">Q</span></em><span style=\"color:black\">3 + 1.5(</span><em><span style=\"color:black\">IQR</span></em><span style=\"color:black\">) = 60 + (1.5)(40) = 120.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The value 300 is greater than 120, so it is a potential outlier. If we delete it and calculate the five values, we get the following values:</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Min = 0</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">Q</span></em><span style=\"color:black\">1 = 20</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>Q</em>3 = 60</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Max = 120</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">We still have 75 percent of the students exercising for 60 minutes or less daily and half of the students exercising between 20 and 60 minutes a day. However, 15 students is a small sample, and the principal should survey more students to be sure of his survey results.</span></p>','','1','2.3.3',NULL,'1','2','1','0','1','2021-06-22 06:04:30','2021-06-22 11:34:30'),(2226,97,'2.4 Box Plots','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong><span style=\"color:black\">Box plots</span></strong><span style=\"color:black\">, also called </span><strong><span style=\"color:black\">box-and-whisker plots </span></strong><span style=\"color:black\">or </span><strong><span style=\"color:black\">box-whisker plots</span></strong><span style=\"color:black\">, give a good graphical image of the concentration of the data. They also show how far the extreme values are from most of the data. As mentioned previously, a box plot is constructed from five values: the minimum value, the first quartile, the median, the third quartile, and the maximum value.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">We use these values to compare how close other data values are to them.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">To construct a box plot, use a horizontal or vertical number line and a rectangular box. The smallest and largest data values label the endpoints of the axis. The first quartile marks one end of the box, and the third quartile marks the other end of the box. Approximately </span><strong><span style=\"color:black\">the middle 50 percent of the data fall inside the box. </span></strong><span style=\"color:black\">The </span><em><span style=\"color:black\">whiskers </span></em><span style=\"color:black\">extend from the ends of the box to the smallest and largest data values. A box plot easily shows the range of a data set, which is the difference between the largest and smallest data values (or the difference between the maximum and minimum). Unless the median, first quartile, and third quartile are the same value, the median will lie inside the box or between the first and third quartiles. The box plot gives a good, quick picture of the data.</span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\">Consider, again, this data set:</span></p>\n\n<p><span style=\"font-size:14px\">1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5</span></p>\n\n<p><span style=\"font-size:14px\">The first quartile is two, the median is seven, and the third quartile is nine. The smallest value is one, and the largest value is 11.5. The following image shows the constructed box plot.</span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238336891623833689.png\" style=\"height:112px; width:416px\" /></span></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:center\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Figure 2.13</strong></span></span></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The two whiskers extend from the first quartile to the smallest value and from the third quartile to the largest value. The median is shown with a dashed line.</span></span></p>\n\n<p>&nbsp;</p>','','1','2.4',NULL,'1','2','1','0','1','2021-06-22 06:06:36','2021-06-22 11:36:36'),(2227,97,'2.4.1 Examples','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Example 2.24</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The following data are the heights of 40 students in a statistics class:</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">59, 60, 61, 62, 62, 63, 63, 64, 64, 64, 65, 65, 65, 65, 65, 65, 65, 65, 65, 66, 66, 67, 67, 68, 68, 69, 70, 70, 70, 70,70, 71, 71, 72, 72, 73, 74, 74, 75, 77.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Construct a box plot with the following properties. Calculator instructions for finding the five number summary</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">follow this example:</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; Minimum value = 59</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; Maximum value = 77</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; </span><em><span style=\"color:black\">Q</span></em><span style=\"color:black\">1: First quartile = 64.5</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; </span><em><span style=\"color:black\">Q</span></em><span style=\"color:black\">2: Second quartile or median = 66</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; </span><em><span style=\"color:black\">Q</span></em><span style=\"color:black\">3: Third quartile = 70</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238341801623834180.png\" style=\"height:138px; width:800px\" /></span></p>\n\n<p style=\"text-align:center\"><span style=\"font-size:14px\"><strong><span style=\"color:#26397f\">Figure 2.14</span></strong></span></p>\n\n<p style=\"text-align:center\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">a. Each quarter has approximately 25 percent of the data.</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">b. The spreads of the four quarters are 64.5 &ndash; 59 = 5.5 (first quarter), 66 &ndash; 64.5 = 1.5 (second quarter), 70 &ndash; 66= 4 (third quarter), and 77 &ndash; 70 = 7 (fourth quarter). So, the second quarter has the smallest spread, and the fourth quarter has the largest spread.</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">c. Range = maximum value &ndash; minimum value = 77 &ndash; 59 = 18.</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">d. Interquartile Range: <em>IQR </em>= <em>Q</em>3 &ndash; <em>Q</em>1 = 70 &ndash; 64.5 = 5.5.</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">e. The interval 59&ndash;65 has more than 25 percent of the data, so it has more data in it than the interval 66&ndash;70, which has 25 percent of the data.</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">f. The middle 50 percent (middle half) of the data has a range of 5.5 inches.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">For some sets of data, some of the largest value, smallest value, first quartile, median, and third quartile may be the same.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">For instance, you might have a data set in which the median and the third quartile are the same. In this case, the diagram would not have a dotted line inside the box displaying the median. The right side of the box would display both the third quartile and the median. For example, if the smallest value and the first quartile were both one, the median and the third quartile were both five, and the largest value was seven, the box plot would look like the following:</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238342221623834222.png\" style=\"height:214px; width:800px\" /></span></p>\n\n<p style=\"text-align:center\"><span style=\"font-size:14px\"><strong><span style=\"color:#26397f\">Figure 2.15</span></strong></span></p>\n\n<p style=\"text-align:center\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">In this case, at least 25 percent of the values are equal to one. Twenty-five percent of the values are between one and five, inclusive. At least 25 percent of the values are equal to five. The top 25 percent of the values fall between five and seven, inclusive.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>Example 2.25</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Test scores for Mr. Ramirez&#39;s class held during the day are as follows:</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">99, 56, 78, 55.5, 32, 90, 80, 81, 56, 59, 45, 77, 84.5, 84, 70, 72, 68, 32, 79, 90.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Test scores for Ms. Park&#39;s class held during the evening are as follows:</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">98, 78, 68, 83, 81, 89, 88, 76, 65, 45, 98, 90, 80, 84.5, 85, 79, 78, 98, 90, 79, 81, 25.5.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">a. Find the smallest and largest values, the median, and the first and third quartile for Mr. Ramirez&#39;s class.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">b. Find the smallest and largest values, the median, and the first and third quartile for Ms. Park&#39;s class.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">c. For each data set, what percentage of the data is between the smallest value and the first quartile? the first quartile and the median? the median and the third quartile? the third quartile and the largest value? What percentage of the data is between the first quartile and the largest value?</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">d. Create a box plot for each set of data. Use one number line for both box plots.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">e. Which box plot has the widest spread for the middle 50 percent of the data, the data between the first and third quartiles? What does this mean for that set of data in comparison to the other set of data?</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution 2.25</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">a. Min = 32</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">Q</span></em><span style=\"color:black\">1 = 56</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">M </span></em><span style=\"color:black\">= 74.5</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">Q</span></em><span style=\"color:black\">3 = 82.5</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Max = 99</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">b. Min = 25.5</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">Q</span></em><span style=\"color:black\">1 = 78</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">M </span></em><span style=\"color:black\">= 81</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">Q</span></em><span style=\"color:black\">3 = 89</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Max = 98</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">c. Mr. Ramirez&#39;s class: There are six data values ranging from 32 to 56: 30 percent. There are six data values ranging from 56 to 74.5: 30 percent. There are five data values ranging from 74.5 to 82.5: 25 percent. There are five data values ranging from 82.5 to 99: 25 percent. There are 16 data values between the first quartile, 56, and the largest value, 99: 75 percent. Ms. Park&rsquo;s class: There are six data values ranging from 25.5 to 78: 27 percent. There are five data values ranging from 78 to the first instance of 81: 23 percent. There are six data values ranging from the second instance of 81 to 89: 27 percent. There are five data values ranging from 90 to 98: 23 percent. There are 17 values between the first quartile, 78, and the largest value, 98: 77 percent.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238342921623834292.png\" style=\"height:296px; width:800px\" /></span></span></p>\n\n<p style=\"text-align:center\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>&nbsp;Figure 2.16</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">d.</span><strong> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </strong></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">e. The first data set has the wider spread for the middle 50 percent of the data. The </span><em><span style=\"color:black\">IQR </span></em><span style=\"color:black\">for the first data set is greater than the </span><em><span style=\"color:black\">IQR </span></em><span style=\"color:black\">for the second set. This means that there is more variability in the middle 50 percent of the first data se</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Example 2.26</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Graph a box-and-whisker plot for the following data values shown:</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">10, 10, 10, 15, 35, 75, 90, 95, 100, 175, 420, 490, 515, 515, 790</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The five numbers used to create a box-and-whisker plot are as follows:</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Min: 10</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">Q</span></em><span style=\"color:black\">1: 15</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Med: 95</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">Q</span></em><span style=\"color:black\">3: 490</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Max: 790</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The following graph shows the box-and-whisker plot.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238345621623834562.png\" style=\"height:158px; width:800px\" /></span></p>\n\n<p style=\"text-align:center\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Figure 2.17</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<h2 style=\"text-align:justify\">&nbsp;</h2>','','1','2.4.1',NULL,'1','2','1','0','1','2021-06-22 06:12:08','2021-06-22 11:42:08'),(2228,97,'2.5 Measures of the Center of the Data','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The </span><em><span style=\"color:black\">center </span></em><span style=\"color:black\">of a data set is also a way of describing location. The two most widely used measures of the </span><em><span style=\"color:black\">center </span></em><span style=\"color:black\">of the data are the </span><strong><span style=\"color:black\">mean </span></strong><span style=\"color:black\">(average) and the </span><strong><span style=\"color:black\">median</span></strong><span style=\"color:black\">. To calculate the </span><strong><span style=\"color:black\">mean weight </span></strong><span style=\"color:black\">of 50 people, add the 50 weights together and divide by 50. To find the </span><strong><span style=\"color:black\">median weight </span></strong><span style=\"color:black\">of the 50 people, order the data and find the number that splits the data into two equal parts. The median is generally a better measure of the center when there are extreme values or outliers because it is not affected by the precise numerical values of the outliers. The mean is the most common measure of the center.</span></span></p>\n\n<p><span style=\"font-size:14px\"><strong>NOTE</strong></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The words </span><em><span style=\"color:black\">mean </span></em><span style=\"color:black\">and </span><em><span style=\"color:black\">average </span></em><span style=\"color:black\">are often used interchangeably. The substitution of one word for the other is common practice. The technical term is </span><em><span style=\"color:black\">arithmetic mean </span></em><span style=\"color:black\">and </span><em><span style=\"color:black\">average </span></em><span style=\"color:black\">is technically a center location. However, in practice among non-statisticians, </span><em><span style=\"color:black\">average </span></em><span style=\"color:black\">is commonly accepted for </span><em><span style=\"color:black\">arithmetic mean</span></em><span style=\"color:black\">.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">When each value in the data set is not unique, the mean can be calculated by multiplying each distinct value by its frequency and then dividing the sum by the total number of data values. The letter used to represent the </span><strong><span style=\"color:black\">sample mean </span></strong><span style=\"color:black\">is an </span><em><span style=\"color:black\">x </span></em><span style=\"color:black\">with a bar over it (pronounced &ldquo;x bar&rdquo;): &nbsp;</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The sample mean is a statistic.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The Greek letter </span><em><span style=\"color:black\">&mu; </span></em><span style=\"color:black\">(pronounced &quot;mew&quot;) represents the </span><strong><span style=\"color:black\">population mean</span></strong><span style=\"color:black\">. The population mean is a parameter. One of the requirements for the </span><strong><span style=\"color:black\">sample mean </span></strong><span style=\"color:black\">to be a good estimate of the </span><strong><span style=\"color:black\">population mean </span></strong><span style=\"color:black\">is for the sample taken to be truly random.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">To see that both ways of calculating the mean are the same, consider the following sample:</span></span></p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">1, 1, 1, 2, 2, 3, 4, 4, 4, 4, 4</span></span></p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238352551623835255.png\" style=\"height:212px; width:800px\" /></span></span></p>\n\n<p><span style=\"font-size:14px\">In the second example, the frequencies are 3(1) + 2(2) + 1(3) + 5(4).</span></p>\n\n<p><span style=\"font-size:14px\">You can quickly find the location of the median by using the expression (<em>n </em>+ 1)/2</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The letter <em>n </em>is the total number of data values in the sample. As discussed earlier, if <em>n </em>is an odd number, the median is the middle value of the ordered data (ordered smallest to largest). If <em>n </em>is an even number, the median is equal to the two middle values added together and divided by two after the data have been ordered. For example, if the total number of data values is 97, then <em>n </em>+ 1</span></p>\n\n<p><span style=\"font-size:14px\">2 = 97 + 1</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">2 = 49. The median is the 49th value in the ordered data. If the total number of data values is 100,</span></p>\n\n<p><span style=\"font-size:14px\">then <em>n </em>+ 1</span></p>\n\n<p><span style=\"font-size:14px\">2 = 100 + 1</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">2 = 50.5. The median occurs midway between the 50th and 51st values. The location of the median and the value of the median are <strong>not </strong>the same. The uppercase letter <em>M </em>is often used to represent the median. </span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The next example illustrates the location of the median and the value of the median.</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Another measure of the center is the mode. The <strong>mode </strong>is the most frequent value. There can be more than one mode in a data set as long as those values have the same frequency and that frequency is the highest. A data set with two modes is called bimodal.</span></p>','','1','2.5',NULL,'1','2','1','1','1','2021-06-16 12:15:08','2021-06-16 17:45:08'),(2229,97,'2.5 Measures of the Center of the Data','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\">The <em>center </em>of a data set is also a way of describing location. The two most widely used measures of the <em>center </em>of the data are the <strong>mean </strong>(average) and the <strong>median</strong>. To calculate the <strong>mean weight </strong>of 50 people, add the 50 weights together and divide by 50. To find the <strong>median weight </strong>of the 50 people, order the data and find the number that splits the data into two equal parts. The median is generally a better measure of the center when there are extreme values or outliers because it is not affected by the precise numerical values of the outliers. The mean is the most common measure of the center.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>NOTE</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The words <em>mean </em>and <em>average </em>are often used interchangeably. The substitution of one word for the other is common practice. The technical term is <em>arithmetic mean </em>and <em>average </em>is technically a center location. However, in practice among non-statisticians, <em>average </em>is commonly accepted for <em>arithmetic mean</em>.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">When each value in the data set is not unique, the mean can be calculated by multiplying each distinct value by its frequency and then dividing the sum by the total number of data values. The letter used to represent the <strong>sample mean </strong>is an <em>x </em>with a bar over it (pronounced &ldquo;x bar&rdquo;): &nbsp;</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The sample mean is a statistic.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The Greek letter <em>&mu; </em>(pronounced &quot;mew&quot;) represents the <strong>population mean</strong>. The population mean is a parameter. One of the requirements for the <strong>sample mean </strong>to be a good estimate of the <strong>population mean </strong>is for the sample taken to be truly random.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">To see that both ways of calculating the mean are the same, consider the following sample:</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">1, 1, 1, 2, 2, 3, 4, 4, 4, 4, 4</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238352551623835255.png\" /></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">In the second example, the frequencies are 3(1) + 2(2) + 1(3) + 5(4).</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">You can quickly find the location of the median by using the expression (<em>n </em>+ 1)/2</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The letter <em>n </em>is the total number of data values in the sample. As discussed earlier, if <em>n </em>is an odd number, the median is the middle value of the ordered data (ordered smallest to largest). If <em>n </em>is an even number, the median is equal to the two middle values added together and divided by two after the data have been ordered. For example, if the total number of data values is 97, then <em>n </em>+ 1</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">2 = 97 + 1</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">2 = 49. The median is the 49th value in the ordered data. If the total number of data values is 100,</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">then <em>n </em>+ 1</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">2 = 100 + 1</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">2 = 50.5. The median occurs midway between the 50th and 51st values. The location of the median and the value of the median are <strong>not </strong>the same. The uppercase letter <em>M </em>is often used to represent the median.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The next example illustrates the location of the median and the value of the median.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Another measure of the center is the mode. The <strong>mode </strong>is the most frequent value. There can be more than one mode in a data set as long as those values have the same frequency and that frequency is the highest. A data set with two modes is called bimodal.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Example 2.27</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Data indicating the number of months a patient with a specific disease lives after taking a new antibody drug are as follows (smallest to largest):</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">3, 4, 8, 8, 10, 11, 12, 13, 14, 15, 15, 16, 16, 17, 17, 18, 21, 22, 22, 24, 24, 25, 26, 26, 27, 27, 29, 29, 31, 32, 33,33, 34, 34, 35, 37, 40, 44, 44, 47</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Calculate the mean and the median.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution 2.27</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The calculation for the mean is</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\"><span class=\"math-tex\">\\(\\bar{x}\\)</span></span>&nbsp;<span style=\"color:black\">= [3 + 4 + (8)(2) + 10 + 11 + 12 + 13 + 14 + (15)(2) + (16)(2) + (17)(2) + 18 + 21 + (22)(2) + (24)(2) + 25 + (26)(2)+ (27)(2) + (29)(2) + 31 + 32 + (33)(2) + (34)(2) + 35 + 37 + 40 + (44)(2) + 47] / 40 = 23.6.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">To find the median, </span><em><span style=\"color:black\">M</span></em><span style=\"color:black\">, first use the formula for the location. The location is</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\"><span class=\"math-tex\">\\(\\frac{n+1}{2}=\\frac{40+1}{2}=20.5\\)</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Start from the smallest value and count up; the median is located between the 20th and 21st values (the two 24s):</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">3, 4, 8, 8, 10, 11, 12, 13, 14, 15, 15, 16, 16, 17, 17, 18, 21, 22, 22, 24, 24, 25, 26, 26, 27, 27, 29, 29, 31, 32, 33, 33, 34, 34, 35, 37, 40, 44, 44, 47</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span class=\"math-tex\">\\(M=\\frac{24+24}{2}\\)</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Example 2.28</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Suppose that in a small town of 50 people, one person earns Rs.5,000,000 per year and the other 49 each earn Rs.30,000. Which is the better measure of the </span><em><span style=\"color:black\">center</span></em><span style=\"color:black\">: the mean or the median?</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution 2.28</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span class=\"math-tex\">\\(\\bar{x}=\\frac{5,000,000+49(30,000)}{50}=129,400\\)</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">M </span></em><span style=\"color:black\">= 30,000</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">There are 49 people who earn Rs.30,000 and one person who earns Rs.5,000,000.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The median is a better measure of the </span><em><span style=\"color:black\">center </span></em><span style=\"color:black\">than the mean because 49 of the values are 30,000 and one is 5,000,000. The 5,000,000 is an outlier. The 30,000 gives us a better sense of the middle of the data.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Another measure of the center is the mode. The <strong>mode </strong>is the most frequent value. There can be more than one mode in a data set as long as those values have the same frequency and that frequency is the highest. A data set with two modes is called bimodal.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>Example 2.29</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Statistics exam scores for 20 students are as follows:</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">50, 53, 59, 59, 63, 63, 72, 72, 72, 72, 72, 76, 78, 81, 83, 84, 84, 84, 90, 93</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Find the mode.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution 2.29</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The most frequent score is 72, which occurs five times. Mode = 72.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Example 2.29</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Five real estate exam scores are 430, 430, 480, 480, 495. The data set is bimodal because the scores 430 and 480 each occur twice.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">When is the mode the best measure of the </span><em><span style=\"color:black\">center</span></em><span style=\"color:black\">? Consider a weight loss program that advertises a mean weight loss of six pounds the first week of the program. The mode might indicate that most people lose two pounds the first week, making the program less appealing.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>The Law of Large Numbers and the Mean</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The Law of Large Numbers says that if you take samples of larger and larger size from any population, then the mean of the sample is very likely to get closer and closer to </span><em><span style=\"color:black\">&mu;</span></em><span style=\"color:black\">. This law is discussed in more detail later in the text.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Sampling Distributions and Statistic of a Sampling Distribution</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">You can think of a sampling distribution as a relative frequency distribution with a great many samples. </span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong><span style=\"color:black\">1: Sampling and Data </span></strong><span style=\"color:black\">for a review of relative frequency. Suppose 30 randomly selected students were asked the number of movies they watched the previous week. The results are in the </span><strong><span style=\"color:black\">relative frequency table </span></strong><span style=\"color:black\">shown below.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238359891623835989.png\" style=\"height:575px; width:800px\" /></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>A relative frequency distribution includes the relative frequencies of a number of samples.</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Recall that a <strong>statistic </strong>is a number calculated from a sample. Statistic examples include the mean, the median, and the mode as well as others. The sample mean <em><span class=\"math-tex\">\\(\\bar{x}\\)</span></em>&nbsp;is an example of a statistic that estimates the population mean <em>&mu;</em>.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Calculating the Mean of Grouped Frequency Tables</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">When only grouped data is available, you do not know the individual data values (we know only intervals and interval frequencies); therefore, you cannot compute an exact mean for the data set. What we must do is estimate the actual mean by calculating the mean of a frequency table. A frequency table is a data representation in which grouped data is displayed along with the corresponding frequencies. To calculate the mean from a grouped frequency table, we can apply the basic definition of mean <img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238370791623837079.png\" style=\"height:25px; width:159px\" /></span>.&nbsp;We simply need to modify the definition to fit within the restrictions of a frequency table.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Since we do not know the individual data values, we can instead find the midpoint of each interval. The midpoint is&nbsp;<img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238371081623837108.png\" style=\"height:26px; width:176px\" />We can now modify the mean definition to be <em>Mean of &nbsp;Frequency Table </em>=&nbsp;</span><span style=\"font-size:10px\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238371661623837166.png\" style=\"height:38px; width:37px\" /></span><span style=\"font-size:14px\">&nbsp;where <em>f </em>= the frequency of the interval, <em>m </em>= the midpoint of the interval, and sigma (&Sigma;) is read as &quot;sigma&quot; and means to sum up. So this formula says that we will sum the products of each midpoint and the corresponding frequency and divide by the sum of all of the frequencies.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">A frequency table displaying Professor Blount&rsquo;s last statistic test is shown. Find the best estimate of the class mean.</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238372181623837218.png\" style=\"height:706px; width:800px\" /></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution 2.31</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; Find the midpoints for all intervals.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238372841623837284.png\" style=\"height:871px; width:800px\" /></span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Calculate the sum of the product of each interval frequency and midpoint. &Sigma; <em>f m</em></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">53.25(1) + 59.5(0) + 65.5(4) + 71.5(4) + 77.5(2) + 83.5(3) + 89.5(4) + 95.5(1) = 1460.25</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&micro;<span class=\"math-tex\">\\(=\\frac{\\sum{fn}}{\\sum{f}}=\\frac{1460.25}{19}=76.86\\)</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\">&nbsp;</p>','','1','2.5',NULL,'1','2','1','0','1','2021-06-22 06:19:10','2021-06-22 11:49:10'),(2230,97,'2.6 Skewness and the Mean, Median, and Mode','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\">Consider the following data set:</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">4, 5, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 9, 10</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">This data set can be represented by the following histogram. Each interval has width 1, and each value is located in the&nbsp;</span></span><span style=\"font-size:14px\"><span style=\"color:black\">middle of an interval.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><br />\n<span style=\"font-size:14px\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238376001623837600.png\" style=\"height:437px; width:800px\" /></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong><span style=\"color:#26397f\">Figure 2.18</span></strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The histogram displays a </span><strong><span style=\"color:black\">symmetrical </span></strong><span style=\"color:black\">distribution of data. A distribution is symmetrical if a vertical line can be drawn at some point in the histogram such that the shape to the left and the right of the vertical line are mirror images of each other.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The mean, the median, and the mode are each seven for these data. </span><strong><span style=\"color:black\">In a perfectly symmetrical distribution, the mean and the median are the same. </span></strong><span style=\"color:black\">This example has one mode (unimodal), and the mode is the same as the mean and median.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">In a symmetrical distribution that has two modes (bimodal), the two modes would be different from the mean and median.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The histogram for the data: 4, 5, 6, 6, 6, 7, 7, 7, 7, 8 is not symmetrical. The right-hand side seems </span><em><span style=\"color:black\">chopped off </span></em><span style=\"color:black\">compared to the left-hand side. A distribution of this type is called </span><strong><span style=\"color:black\">skewed to the left </span></strong><span style=\"color:black\">because it is pulled out to the left. A skewed left distribution has more high values.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238376511623837651.png\" style=\"height:579px; width:800px\" /></span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong><span style=\"color:#26397f\">Figure 2.19</span></strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The mean is 6.3, the median is 6.5, and the mode is seven. </span><strong><span style=\"color:black\">Notice that the mean is less than the median, and they are both less than the mode</span></strong><span style=\"color:black\">. The mean and the median both reflect the skewing, but the mean reflects it more so. The mean is pulled toward the tail in a skewed distribution.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The histogram for the data: 6, 7, 7, 7, 7, 8, 8, 8, 9, 10 is also not symmetrical. It is </span><strong><span style=\"color:black\">skewed to the right</span></strong><span style=\"color:black\">. A skewed right distribution has more low values.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238377051623837705.png\" style=\"height:575px; width:800px\" /></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong><span style=\"color:#26397f\">Figure 2.20</span></strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The mean is 7.7, the median is 7.5, and the mode is seven. Of the three statistics, </span><strong><span style=\"color:black\">the mean is the largest, while the mode is the smallest</span></strong><span style=\"color:black\">. Again, the mean reflects the skewing the most.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">To summarize, generally if the distribution of data is skewed to the left, the mean is less than the median, which is often less than the mode. If the distribution of data is skewed to the right, the mode is often less than the median, which is less than the mean.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Skewness and symmetry become important when we discuss probability distributions in later chapters.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>','','1','2.6',NULL,'1','2','1','0','1','2021-06-22 06:22:30','2021-06-22 11:52:30'),(2231,97,'2.6.1  Examples','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong><span style=\"color:#002060\">Example 2.32</span></strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Statistics are used to compare and sometimes identify authors. The following lists show a simple random sample <span style=\"color:black\">that compares the letter counts for three authors.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Terry: 7, 9, 3, 3, 3, 4, 1, 3, 2, 2</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Davis: 3, 3, 3, 4, 1, 4, 3, 2, 3, 1</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Maris: 2, 3, 4, 4, 4, 6, 6, 6, 8, 3</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">a. Make a dot plot for the three authors and compare the shapes.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">b. Calculate the mean for each.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">c. Calculate the median for each.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">d. Describe any pattern you notice between the shape and the measures of center.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution 2.32</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238379651623837965.png\" style=\"height:238px; width:800px\" /></strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><strong><span style=\"color:#26397f\">Figure 2.21 </span></strong><span style=\"color:black\">Terry&rsquo;s distribution has a right (positive) skew.</span></span></p>\n\n<p>&nbsp;</p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238379961623837996.png\" style=\"height:196px; width:800px\" /></span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><strong><span style=\"color:#26397f\">Figure 2.22 </span></strong><span style=\"color:black\">Davis&rsquo;s distribution has a left (negative) skew.</span></span></p>\n\n<p>&nbsp;</p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238380311623838031.png\" style=\"height:209px; width:800px\" /></span></span></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong><span style=\"color:#000000\">Figure 2.23 Maris&rsquo;s distribution is symmetrically shaped.</span></strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">b. Terry&rsquo;s mean is 3.7, Davis&rsquo;s mean is 2.7, and Maris&rsquo;s mean is 4.6.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">c. Terry&rsquo;s median is 3, Davis&rsquo;s median is 3, and Maris&rsquo;s median is four. It would be helpful to manually calculate these descriptive statistics, using the given data sets and then compare to the graphs.</span> d. It appears that the median is always closest to the high point (the mode), while the mean tends to be farther out on the tail. In a symmetrical distribution, the mean and the median are both centrally located close to the high point of the distribution.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238380691623838069.png\" style=\"height:445px; width:800px\" /></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><strong>Figure2.25</strong></span></p>','','1','2.6.1',NULL,'1','2','1','0','1','2021-06-22 06:26:27','2021-06-22 11:56:27'),(2232,97,'2.7 Measures of the Spread of the Data','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">An important characteristic of any set of data is the variation in the data. In some data sets, the data values are concentrated closely near the mean; in other data sets, the data values are more widely spread out from the mean. The most common measure of variation, or spread, is the standard deviation. The </span><strong><span style=\"color:black\">standard deviation </span></strong><span style=\"color:black\">is a number that measures how far data values are from their mean.</span></span></p>','','1','2.7',NULL,'1','2','1','0','1','2021-06-22 06:27:09','2021-06-22 11:57:09'),(2233,97,'2.7.1 The standard deviation','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; provides a numerical measure of the overall amount of variation in a data set and</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; can be used to determine whether a particular data value is close to or far from the mean.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>The standard deviation provides a measure of the overall variation in a data set.</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The standard deviation is always positive or zero. The standard deviation is small when all the data are concentrated close to the mean, exhibiting little variation or spread. The standard deviation is larger when the data values are more spread out from the mean, exhibiting more variation.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Suppose that we are studying the amount of time customers wait in line at the checkout at Supermarket A and Supermarket&nbsp;B. The average wait time at both supermarkets is five minutes. At Supermarket A, the standard deviation for the wait time is two minutes; at Supermarket B, the standard deviation for the wait time is four minutes.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Because Supermarket B has a higher standard deviation, we know that there is more variation in the wait times at Supermarket B. Overall, wait times at Supermarket B are more spread out from the average whereas wait times at Supermarket A are more concentrated near the average.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>The standard deviation can be used to determine whether a data value is close to or far from the mean.</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Suppose that both Roja and Binu shop at Supermarket A. Roja waits at the checkout counter for seven minutes, and Binu waits for one minute. At Supermarket A, the mean waiting time is five minutes, and the standard deviation is two minutes.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The standard deviation can be used to determine whether a data value is close to or far from the mean. A </span><em><span style=\"color:black\">z</span></em><span style=\"color:black\">-score is a standardized score that lets us compare data sets. It tells us how many standard deviations a data value is from the mean and is calculated as the ratio of the difference in a particular score and the population mean to the population standard deviation.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">We can use the given information to create the table below.</span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238383151623838315.png\" style=\"height:152px; width:800px\" /></span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\">Since Roja and Binu only shop at Supermarket A, we can ignore the row for Supermarket B.</span></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">We need the values from the first row to determine the number of standard deviations above or below the mean each individual wait time is; we can do so by calculating two different <em>z</em>-scores.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p><span style=\"font-size:14px\">Rosa waited for seven minutes, so the <em>z</em>-score representing this deviation from the population mean may be calculated as&nbsp;<span class=\"math-tex\">\\(Z=\\frac{x-\\mu}{\\sigma}=\\frac{7-5}{2}=1\\)</span>&nbsp;.</span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\">The<em> z</em>-score of one tells us that Rosa&rsquo;s wait time is one standard deviation above the mean wait time of five minutes.</span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\">Binh waited for one minute, so the <em>z</em>-score representing this deviation from the population mean may be calculated as&nbsp;<span class=\"math-tex\">\\(Z=\\frac{x-\\mu}{\\sigma}=\\frac{1-5}{2}=-2\\)</span>&nbsp;.</span></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The <em>z</em>-score of &minus;2 tells us that Binh&rsquo;s wait time is two standard deviations below the mean wait time of five minutes.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">A data value that is two standard deviations from the average is just on the borderline for what many statisticians would consider to be far from the average. Considering data to be far from the mean if they are more than two standard deviations away is more of an approximate <em>rule of thumb </em>than a rigid rule. In general, the shape of the distribution of the data affects how much of the data is farther away than two standard deviations. You will learn more about this in later chapters.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The number line may help you understand standard deviation. If we were to put five and seven on a number line, seven is to the right of five. We say, then, that seven is <strong>one </strong>standard deviation to the <strong>right </strong>of five because 5 + (1)(2) = 7.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p><span style=\"font-size:14px\">If one were also part of the data set, then one is two standard deviations to the left of five because 5 + (&ndash;2)(2) = 1.</span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238386711623838671.png\" style=\"height:77px; width:800px\" /></span></p>\n\n<p>&nbsp;</p>\n\n<h3>&nbsp;</h3>\n\n<p><span style=\"font-size:14px\"><strong><span style=\"color:#26397f\">Figure 2.26</span></strong></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">&bull; In general, a </span><strong><span style=\"color:black\">value = mean + (#ofSTDEV)(standard deviation)</span></strong></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">&bull; where #ofSTDEVs = the number of standard deviations</span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">&bull; #ofSTDEV does not need to be an integer</span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">&bull; One is </span><strong><span style=\"color:black\">two standard deviations less than the mean </span></strong><span style=\"color:black\">of five because 1 = 5 + (&ndash;2)(2).</span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">The equation </span><strong><span style=\"color:black\">value = mean + (#ofSTDEVs)(standard deviation) </span></strong><span style=\"color:black\">can be expressed for a sample and for a population as&nbsp;follows:</span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">&bull; </span><strong><span style=\"color:black\">Sample</span></strong><em> </em><span class=\"math-tex\">\\(x =\\bar{x}+\\)</span>( # <em>o f STDEV</em>)(<em>s</em>)</span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">The lowercase letter </span><em><span style=\"color:black\">s </span></em><span style=\"color:black\">represents the sample standard deviation and the Greek letter </span><em><span style=\"color:black\">&sigma; </span></em><span style=\"color:black\">(lower case) represents the population standard deviation. The symbol </span><span class=\"math-tex\">\\(\\bar{x}\\)</span>&nbsp;<span style=\"color:black\">is the sample mean, and the Greek symbol </span><em><span style=\"color:black\">&mu; </span></em><span style=\"color:black\">is the population mean.</span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Calculating the Standard Deviation</strong></span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">If </span><em><span style=\"color:black\">x </span></em><span style=\"color:black\">is a number, then the difference </span><em><span style=\"color:black\">x</span></em><em><span style=\"color:black\">&nbsp;mean </span></em><span style=\"color:black\">is called its </span><strong><span style=\"color:black\">deviation</span></strong><span style=\"color:black\">. In a data set, there are as many deviations as there are items in the data set. The deviations are used to calculate the standard deviation. If the numbers belong to a population, in symbols, a deviation is </span><em><span style=\"color:black\">x </span></em><span style=\"color:black\">&ndash; </span><em><span style=\"color:black\">&mu;</span></em><span style=\"color:black\">. For sample data, in symbols, a deviation is </span><em><span style=\"color:black\">x </span></em><span style=\"color:black\">&ndash;&nbsp;</span><span class=\"math-tex\">\\(\\bar{x}\\)</span>.</span></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The procedure to calculate the standard deviation depends on whether the numbers are the entire population or are data from a sample. The calculations are similar but not identical. Therefore, the symbol used to represent the standard deviation depends on whether it is calculated from a population or a sample. The lowercase letter </span><em><span style=\"color:black\">s </span></em><span style=\"color:black\">represents the sample standard deviation and the Greek letter </span><em><span style=\"color:black\">&sigma; </span></em><span style=\"color:black\">(lowercase sigma) represents the population standard deviation. If the sample has the same characteristics as the population, then </span><em><span style=\"color:black\">s </span></em><span style=\"color:black\">should be a good estimate of </span><em><span style=\"color:black\">&sigma;</span></em><span style=\"color:black\">.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">To calculate the standard deviation, we need to calculate the variance first. The </span><strong><span style=\"color:black\">variance </span></strong><span style=\"color:black\">is the </span><strong><span style=\"color:black\">average of the squares of the deviations </span></strong><span style=\"color:black\">(the </span><em><span style=\"color:black\">x&nbsp;</span></em><span class=\"math-tex\">\\(-\\bar{x}\\)</span>&nbsp;</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">values for a sample or the </span><em><span style=\"color:black\">x </span></em><span style=\"color:black\">&ndash; </span><em><span style=\"color:black\">&mu; </span></em><span style=\"color:black\">values for a population). The symbol </span><em><span style=\"color:black\">&sigma;</span></em><sup><span style=\"color:black\">2</span></sup><span style=\"color:black\"> represents the population variance; the population standard deviation </span><em><span style=\"color:black\">&sigma; </span></em><span style=\"color:black\">is the square root of the population variance. The symbol </span><em><span style=\"color:black\">s</span></em><span style=\"color:black\">2 represents the&nbsp;sample variance; the sample standard deviation </span><em><span style=\"color:black\">s </span></em><span style=\"color:black\">is the square root of the sample variance. You can think of the standard deviation as a special average of the deviations.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">If the numbers come from a census of the entire </span><strong><span style=\"color:black\">population </span></strong><span style=\"color:black\">and not a sample, when we calculate the average of the squared deviations to find the variance, we divide by </span><em><span style=\"color:black\">N</span></em><span style=\"color:black\">, the number of items in the population. If the data are from a </span><strong><span style=\"color:black\">sample </span></strong><span style=\"color:black\">rather than a population, when we calculate the average of the squared deviations, we divide by <strong><em>n </em></strong></span><strong><span style=\"color:black\">&ndash; 1</span></strong><span style=\"color:black\">, one less than the number of&nbsp; items in the sample.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#000000\">Formulas for the Sample Standard Deviation</span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#26397f\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238390581623839058.png\" style=\"height:81px; width:400px\" /></span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\">&bull; For the sample standard deviation, the denominator is <em>n</em>&minus;; that is, the sample size minus 1.</span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#000000\">Formulas for the Population Standard Deviation</span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#26397f\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238390941623839094.png\" style=\"height:54px; width:400px\" /></span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">&bull; For the population standard deviation, the denominator is </span><em><span style=\"color:black\">N</span></em><span style=\"color:black\">, the number of items in the population.</span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">In these formulas, </span><em><span style=\"color:black\">f </span></em><span style=\"color:black\">represents the frequency with which a value appears. For example, if a value appears once, </span><em><span style=\"color:black\">f </span></em><span style=\"color:black\">is one. If a value appears three times in the data set or population, </span><em><span style=\"color:black\">f </span></em><span style=\"color:black\">is three.</span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Types of Variability in Samples</strong></span></span></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">When researchers study a population, they often use a sample, either for convenience or because it is not possible to access the entire population. </span><em><span style=\"color:black\">Variability </span></em><span style=\"color:black\">is the term used to describe the differences that may occur in these outcomes. Common types of variability include the following:</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">&bull; Observational or measurement variability</span></span></p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">&bull; Natural variability</span></span></p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">&bull; Induced variability</span></span></p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">&bull; Sample variability</span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">Here are some examples to describe each type of variability:</span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><strong><span style=\"color:black\">Example 1: Measurement variability</span></strong></span></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Measurement variability occurs when there are differences in the instruments used to measure or in the people using those instruments. If we are gathering data on how long it takes for a ball to drop from a height by having students measure the time of the drop with a stopwatch, we may experience measurement variability if the two stopwatches used were made by different manufacturers. For example, one stopwatch measures to the nearest second, whereas the other one measures to the nearest tenth of a second. We also may experience measurement variability because two different people are gathering the data. Their reaction times in pressing the button on the stopwatch may differ; thus, the outcomes will vary accordingly. The differences in outcomes may be affected by measurement variability.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><strong><span style=\"color:black\">Example 2: Natural variability</span></strong></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">Natural variability arises from the differences that naturally occur because members of a population differ from each other.</span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">For example, if we have two identical corn plants and we expose both plants to the same amount of water and sunlight, they may still grow at different rates simply because they are two different corn plants. The difference in outcomes may be explained by natural variability.</span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><strong><span style=\"color:black\">Example 3: Induced variability</span></strong></span></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Induced variability is the counterpart to natural variability. This occurs because we have &nbsp;artificially induced an element of variation that, by definition, was not present naturally. For example, we assign people to two different groups to study memory, and we induce a variable in one group by limiting the amount of sleep they get. The difference in outcomes may&nbsp;be affected by induced variability.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><strong><span style=\"color:black\">Example 4: Sample variability</span></strong></span></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Sample variability occurs when multiple random samples are taken from the same population. For example, if I conduct four surveys of 50 people randomly selected from a given population, the differences in outcomes may be affected by sample variability.</span></span></p>','','1','2.7.1',NULL,'1','2','1','0','1','2021-06-23 05:02:27','2021-06-23 10:32:27'),(2234,97,'2.7.2 Sampling Variability of a Statistic','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The statistic of a sampling distribution was discussed in </span><strong><span style=\"color:#26397f\">Descriptive Statistics: Measures of the Center of the Data</span></strong><span style=\"color:black\">. How much the statistic varies from one sample to another is known as the </span><strong><span style=\"color:black\">sampling variability of a statistic</span></strong><span style=\"color:black\">. You typically measure the sampling variability of a statistic by its standard error. The </span><strong><span style=\"color:black\">standard error of the mean </span></strong><span style=\"color:black\">is an example of a standard error. The standard error is the standard deviation of the sampling distribution. In other words, it is the average standard deviation that results from repeated sampling. You will cover the standard error of the mean in the chapter </span><strong><span style=\"color:#26397f\">The Central Limit Theorem </span></strong><span style=\"color:black\">(not now). The notation for the standard error of the mean is </span><em><span style=\"color:black\">&sigma; n </span></em><span style=\"color:black\">, where </span><em><span style=\"color:black\">&sigma; </span></em><span style=\"color:black\">is the standard deviation of the population and </span><em><span style=\"color:black\">n </span></em><span style=\"color:black\">is the size of the sample.</span></span></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>EXAMPLE&nbsp;2.33</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">In a fifth-grade class, the teacher was interested in the average age and the sample standard deviation of the ages of her students. The following data are the ages for a SAMPLE of&nbsp;<em>n</em>&nbsp;= 20 fifth-grade students. The ages are rounded to the nearest half year.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">9, 9.5, 9.5, 10, 10, 10, 10, 10.5, 10.5, 10.5, 10.5, 11, 11, 11, 11, 11, 11, 11.5, 11.5, 11.5</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span class=\"math-tex\">\\(\\bar{x}\\)</span></em>=(9&nbsp;+&nbsp;9.5(2)&nbsp;+&nbsp;10(4)&nbsp;+&nbsp;10.5(4)&nbsp;+&nbsp;11(6)&nbsp;</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&nbsp; &nbsp; &nbsp;+&nbsp;11.5(3)/2) = 10.525</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The average age is 10.53 years, rounded to two places.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The variance may be calculated by using a table. Then the standard deviation is calculated by taking the square root of the variance. We will explain the parts of the table after calculating&nbsp;<em>s</em>.</span></p>\n\n<p>&nbsp;</p>\n\n<p>&nbsp;</p>\n\n<table cellspacing=\"0\" class=\"Table\" style=\"border-collapse:collapse; border:none; width:900px\">\n	<thead>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom; width:60px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Data</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom; width:96px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Frequency</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom; width:150px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Deviations</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom; width:150px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong><em>Deviations</em><sup>2</sup></strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom; width:174px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>(Frequency)(<em>Deviations</em><sup>2</sup>)</strong></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; width:60px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><em>x</em></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; width:96px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><em>f</em></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom; width:150px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span class=\"math-tex\">\\((x-\\bar{x})\\)</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom; width:150px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span class=\"math-tex\">\\((x-\\bar{x})^2\\)</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom; width:174px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span class=\"math-tex\">\\((f)(x-\\bar{x})^2\\)</span></span></div>\n			</td>\n		</tr>\n	</thead>\n</table>\n\n<div style=\"text-align:center\">&nbsp;</div>\n\n<table cellspacing=\"0\" class=\"Table\" style=\"border-collapse:collapse; border:none; width:900px\">\n	<tbody>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; width:60px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">9</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; width:96px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">1</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; width:150px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">9 &ndash; 10.525 = &ndash;1.525</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; width:150px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">(&ndash;1.525)<sup>2</sup>&nbsp;= 2.325625</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; width:174px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">1 &times; 2.325625 = 2.325625</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; width:60px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">9.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:96px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">2</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:150px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">9.5 &ndash; 10.525 = &ndash;1.025</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:150px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">(&ndash;1.025)<sup>2</sup>&nbsp;= 1.050625</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:174px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">2 &times; 1.050625 = 2.101250</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; width:60px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">10</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:96px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">4</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:150px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">10 &ndash; 10.525 = &ndash;.525</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:150px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">(&ndash;.525)<sup>2</sup>&nbsp;= .275625</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:174px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">4 &times; .275625 = 1.1025</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; width:60px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">10.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:96px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">4</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:150px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">10.5 &ndash; 10.525 = &ndash;.025</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:150px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">(&ndash;.025)<sup>2</sup>&nbsp;= .000625</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:174px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">4 &times; .000625 = .0025</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; width:60px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">11</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:96px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">6</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:150px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">11 &ndash; 10.525 = .475</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:150px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">(.475)<sup>2</sup>&nbsp;= .225625</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:174px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">6 &times; .225625 = 1.35375</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; width:60px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">11.5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:96px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">3</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:150px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">11.5 &ndash; 10.525 = .975</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:150px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">(.975)<sup>2</sup>&nbsp;= .950625</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:174px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">3 &times; .950625 = 2.851875</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; width:60px\">\n			<div style=\"text-align:center\">&nbsp;</div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:96px\">\n			<div style=\"text-align:center\">&nbsp;</div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:150px\">\n			<div style=\"text-align:center\">&nbsp;</div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:150px\">\n			<div style=\"text-align:center\">&nbsp;</div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; width:174px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">The total is 9.7375.</span></div>\n			</td>\n		</tr>\n	</tbody>\n</table>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The last column simply multiplies each squared deviation by the frequency for the corresponding data value.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The sample variance,&nbsp;<em>s</em><sup>2</sup>, is equal to the sum of the last column (9.7375) divided by the total number of data values minus one (20 &ndash; 1):</span><span style=\"font-size:14px\"><span class=\"math-tex\">\\(s^2=\\frac{9.7375}{20-1}=.5125\\)</span>&nbsp;.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The&nbsp;<strong>sample standard deviation</strong>&nbsp;<em>s</em>&nbsp;is equal to the square root of the sample variance&nbsp;<span class=\"math-tex\">\\(s=\\sqrt{.5125}=.715891\\)</span>.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">which is rounded to two decimal places,&nbsp;<em>s</em>&nbsp;= .72.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Typically, you do the calculation for the standard deviation on your calculator or computer. The intermediate results are not rounded. This is done for accuracy.</span></p>\n\n<p>&nbsp;</p>\n\n<ul>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\">For the following problems, recall that&nbsp;<strong>value = mean + (#ofSTDEVs)(standard deviation)</strong>. Verify the mean and standard deviation on a calculator or computer. Note that these formulas are derived by algebraically manipulating the&nbsp;<em>z</em>-score formulas, given either parameters or statistics.</span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\">For a sample:&nbsp;<span class=\"math-tex\">\\(x =\\bar{x}+\\)</span>(#ofSTDEVs)(<em>s</em>)</span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\">For a population:&nbsp;<em>x</em>&nbsp;=&nbsp;<em>&mu;</em>&nbsp;+ (#ofSTDEVs)(<em>&sigma;</em>)</span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\">For this example, use&nbsp;<span class=\"math-tex\">\\(x =\\bar{x}+\\)</span>&nbsp;(#ofSTDEVs)(<em>s</em>) because the data is from a sample</span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\">Verify the mean and standard deviation on your calculator or computer.</span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\">Find the value that is one standard deviation above the mean. Find&nbsp;<span class=\"math-tex\">\\((\\bar{x}+1s)\\)</span>&nbsp;.</span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\">Find the value that is two standard deviations below the mean. Find&nbsp;<span class=\"math-tex\">\\((\\bar{x}-2s)\\)</span></span></p>\n	</li>\n</ul>\n\n<p>&nbsp;</p>\n\n<ol style=\"list-style-type:lower-alpha\">\n	<li>\n	<p><span style=\"font-size:14px\">Find the values that are 1.5 standard deviations&nbsp;<strong>from</strong>&nbsp;(below and above) the mean.</span></p>\n	</li>\n</ol>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong><span style=\"background-color:white\">Explanation of the standard deviation calculation shown in the table</span></strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">The deviations show how spread out the data are about the mean. The data value 11.5 is farther from the mean than is the data value 11, which is indicated by the deviations .97 and .47. A positive deviation occurs when the data value is greater than the mean, whereas a negative deviation occurs when the data value is less than the mean. The deviation is &ndash;1.525 for the data value nine.&nbsp;<strong>If you add the deviations, the sum is always zero</strong>. We can sum the products of the frequencies and deviations to show that the sum of the deviations is always zero.</span></span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">1(&minus;1.525)+2(&minus;1.025)+4(&minus;.525)+4(&minus;.025)</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&nbsp; +6(.475)+3(.975)=0</span></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">For&nbsp;<a href=\"https://openstax.org/books/statistics/pages/2-7-measures-of-the-spread-of-the-data#element-655\" style=\"color:blue; text-decoration:underline\">Example 2.33</a>, there are&nbsp;<em>n</em>&nbsp;= 20 deviations. So you cannot simply add the deviations to get the spread of the data. By squaring the deviations, you make them positive numbers, and the sum will also be positive. The variance, then, is the average squared deviation.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">The variance is a squared measure and does not have the same units as the data. Taking the square root solves the problem. The standard deviation measures the spread in the same units as the data.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">Notice that instead of dividing by&nbsp;<em>n</em>&nbsp;= 20, the calculation divided by&nbsp;<em>n</em>&nbsp;&ndash; 1 = 20 &ndash; 1 = 19 because the data is a sample. For the&nbsp;<strong>sample</strong>&nbsp;variance, we divide by the sample size minus one (<em>n</em>&nbsp;&ndash; 1). Why not divide by&nbsp;<em>n</em>? The answer has to do with the population variance.&nbsp;<strong>The sample variance is an estimate of the population variance.</strong>&nbsp;Based on the theoretical mathematics that lies behind these calculations, dividing by (<em>n</em>&nbsp;&ndash; 1) gives a better estimate of the population variance.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>NOTE</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Your concentration should be on what the standard deviation tells us about the data. The standard deviation is a number that measures how far the data are spread from the mean. Let a calculator or computer do the arithmetic.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">The standard deviation,&nbsp;<em>s</em>&nbsp;or&nbsp;<em>&sigma;</em>, is either zero or larger than zero. Describing the data with reference to the spread is called&nbsp;<em>variability</em>. The variability in data depends on the method by which the outcomes are obtained, for example, by measuring or by random sampling. When the standard deviation is zero, there is no spread; that is, all the data values are equal to each other. The standard deviation is small when all the data are concentrated close to the mean and larger when the data values show more variation from the mean. When the standard deviation is a lot larger than zero, the data values are very spread out about the mean; outliers can make&nbsp;<em>s</em>&nbsp;or&nbsp;<em>&sigma;</em>&nbsp;very large.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">The standard deviation, when first presented, can seem unclear. By graphing your data, you can get a better&nbsp;<em>feel</em>&nbsp;for the deviations and the standard deviation. You will find that in symmetrical distributions, the standard deviation can be very helpful, but in skewed distributions, the standard deviation may not be much help. The reason is that the two sides of a skewed distribution have different spreads. In a skewed distribution, it is better to look at the first quartile, the median, the third quartile, the smallest value, and the largest value. Because numbers can be confusing,&nbsp;<strong>always graph your data</strong>. Display your data in a histogram or a box plot.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>EXAMPLE&nbsp;2.34</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Use the following data (first exam scores) from Susan Dean&#39;s spring precalculus class:</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">33, 42, 49, 49, 53, 55, 55, 61, 63, 67, 68, 68, 69, 69, 72, 73, 74, 78, 80, 83, 88, 88, 88, 90, 92, 94, 94, 94, 94, 96, 100</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<ol style=\"list-style-type:lower-alpha\">\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Create a chart containing the data, frequencies, relative frequencies, and cumulative relative frequencies to three decimal places.</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Calculate the following to one decimal place using a TI-83+ or TI-84 calculator:</span></span></p>\n\n	<ol style=\"list-style-type:lower-roman\">\n		<li>\n		<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The sample mean</span></span></p>\n		</li>\n		<li>\n		<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The sample standard deviation</span></span></p>\n		</li>\n		<li>\n		<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The median</span></span></p>\n		</li>\n		<li>\n		<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The first quartile</span></span></p>\n		</li>\n		<li>\n		<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The third quartile</span></span></p>\n		</li>\n		<li>\n		<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\"><em>IQR</em></span></span></p>\n		</li>\n	</ol>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Construct a box plot and a histogram on the same set of axes. Make comments about the box plot, the histogram, and the chart.</span></span></p>\n	</li>\n</ol>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong><span style=\"color:#2e74b5\"><span style=\"color:black\">Solution&nbsp;1</span></span></strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<ol style=\"list-style-type:lower-alpha\">\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">See&nbsp;<a href=\"https://openstax.org/books/statistics/pages/2-7-measures-of-the-spread-of-the-data#id6947804\" style=\"color:blue; text-decoration:underline\"><span style=\"color:black\">Table 2.33</span></a>.</span></span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">Entering the data values into a list in your graphing calculator and then selecting Stat, Calc, and 1-Var Stats will produce the one-variable statistics you need.</span></span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">The&nbsp;<em>x</em>-axis goes from 32.5 to 100.5; the&nbsp;<em>y</em>-axis goes from &ndash;2.4 to 15 for the histogram. The number of intervals is 5, so the width of an interval is (100.5 &ndash; 32.5) divided by 5, equal to 13.6. Endpoints of the intervals are as follows: the starting point is 32.5, 32.5 + 13.6 = 46.1, 46.1 + 13.6 = 59.7, 59.7 + 13.6 = 73.3, 73.3 + 13.6 = 86.9, 86.9 + 13.6 = 100.5 = the ending value; no data values fall on an interval boundary.</span></span></p>\n	</li>\n</ol>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238402971623840297.jpg\" style=\"height:523px; width:800px\" /></span></span></p>\n\n<p><strong><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:#424242\">Figure&nbsp;</span><span style=\"color:#424242\">2.27</span> </span></span></strong></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">The long left whisker in the box plot is reflected in the left side of the histogram. The spread of the exam scores in the lower 50 percent is greater (73 &ndash; 33 = 40) than the spread in the upper 50 percent (100 &ndash; 73 = 27). The histogram, box plot, and chart all reflect this. There are a substantial number of A and B grades (80s, 90s, and 100). The histogram clearly shows this. The box plot shows us that the middle 50 percent of the exam scores (<em>IQR</em>&nbsp;= 29) are Ds, Cs, and Bs. The box plot also shows us that the lower 25 percent of the exam scores are Ds and Fs.</span></span></span></p>\n\n<p>&nbsp;</p>\n\n<table cellspacing=\"0\" class=\"Table\" style=\"background:white; border-collapse:collapse; border:none; width:900px\">\n	<caption>\n	<p><span style=\"font-size:14px\"><strong>Table 2.33</strong></span></p>\n	</caption>\n	<thead>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; height:15px; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong><span style=\"color:#424242\">Data</span></strong></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:15px; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong><span style=\"color:#424242\">Frequency</span></strong></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:15px; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong><span style=\"color:#424242\">Relative Frequency</span></strong></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:15px; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong><span style=\"color:#424242\">Cumulative Relative Frequency</span></strong></span></div>\n			</td>\n		</tr>\n	</thead>\n	<tbody>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">33</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">1</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.032</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.032</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:15px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">42</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:15px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">1</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:15px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.032</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:15px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.064</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">49</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">2</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.065</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.129</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:15px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">53</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:15px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">1</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:15px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.032</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:15px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.161</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">55</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">2</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.065</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.226</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">61</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">1</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.032</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.258</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">63</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">1</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.032</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.290</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">67</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">1</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.032</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.322</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">68</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">2</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.065</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.387</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">69</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">2</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.065</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.452</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">72</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">1</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.032</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.484</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">73</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">1</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.032</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.516</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">74</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">1</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.032</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.548</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">78</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">1</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.032</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.580</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">80</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">1</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.032</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.612</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">83</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">1</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.032</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.644</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">88</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">3</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.097</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.741</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">90</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">1</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.032</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.773</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">92</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">1</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.032</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.805</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">94</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">4</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.129</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.934</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">96</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">1</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.032</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.966</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">100</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">1</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.032</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.998 (Why isn&#39;t this value 1?)</span></span></div>\n			</td>\n		</tr>\n	</tbody>\n</table>\n\n<p>&nbsp;</p>\n\n<ul>\n</ul>','','1','2.7.2',NULL,'1','2','1','0','1','2021-06-23 05:17:26','2021-06-23 10:47:26'),(2235,97,'2.7.3 Standard deviation of Grouped Frequency Tables','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\">Recall that for grouped data we do not know individual data values, so we cannot describe the typical value of the data with precision. In other words, we cannot find the exact mean, median, or mode. We can, however, determine the best estimate of the measures of center by finding the mean of the grouped data with the formula&nbsp; Mean &nbsp;of &nbsp;Frequency&nbsp;<span class=\"math-tex\">\\(Table=\\frac{\\sum{fm}}{\\sum{f}}\\)</span><span style=\"background-color:white\"><span style=\"color:black\">, where&nbsp;f=&nbsp;interval frequencies and&nbsp;<em>m</em>&nbsp;= interval midpoints.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">Just as we could not find the exact mean, neither can we find the exact standard deviation. Remember that standard deviation describes numerically the expected deviation a data value has from the mean. In simple English, the standard deviation allows us to compare how&nbsp;<em>unusual</em>&nbsp;individual data are when compared to the mean.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>EXAMPLE&nbsp;2.35</strong></span></span></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Find the standard deviation for the data in&nbsp;</span><a href=\"https://openstax.org/books/statistics/pages/2-7-measures-of-the-spread-of-the-data#fs-idm67330768\" style=\"color:blue; text-decoration:underline\"><span style=\"color:#027eb5\">Table 2.34</span></a><span style=\"color:#424242\">.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<table cellspacing=\"0\" class=\"Table\" style=\"border-collapse:collapse; border:none; width:900px\">\n	<thead>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; height:49px; vertical-align:bottom; width:72px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Class</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:49px; vertical-align:bottom; width:123px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Frequency,&nbsp;<em>f</em></strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:49px; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Midpoint,&nbsp;<em>m</em></strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:49px; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><em><strong>m</strong></em><strong><sup>2</sup></strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:49px; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span class=\"math-tex\">\\(\\bar{x}^2\\)</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:49px; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><em><strong>fm</strong></em><strong><sup>2</sup></strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:49px; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Standard Deviation</strong></span></div>\n			</td>\n		</tr>\n	</thead>\n	<tbody>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:30px; width:72px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">0&ndash;2</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px; width:123px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">1</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">1</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">1</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">7.58</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">1</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">3.5</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:30px; width:72px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">3&ndash;5</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px; width:123px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">6</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">4</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">16</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">7.58</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">96</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">3.5</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:30px; width:72px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">6&ndash;8</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px; width:123px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">10</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">7</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">49</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">7.58</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">490</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">3.5</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:30px; width:72px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">9&ndash;11</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px; width:123px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">7</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">10</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">100</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">7.58</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">700</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:30px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">3.5</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:49px; width:72px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">12&ndash;14</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px; width:123px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">0</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">13</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">169</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">7.58</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">0</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">3.5</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:49px; width:72px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">15&ndash;17</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px; width:123px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">2</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">16</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">256</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">7.58</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">512</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:49px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">3.5</span></div>\n			</td>\n		</tr>\n	</tbody>\n</table>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">For this data set, we have the mean,&nbsp;</span><span class=\"math-tex\">\\(\\bar{x}=7.58\\)</span>&nbsp;<span style=\"color:black\">and the standard deviation,&nbsp;<em>s<sub>x</sub></em>&nbsp;= 3.5. This means that a randomly selected data value would be expected to be 3.5 units from the mean. If we look at the first class, we see that the class midpoint is equal to one. This is almost two full standard deviations from the mean since 7.58 &ndash; 3.5 &ndash; 3.5 = .58. While the formula for calculating the standard deviation is not complicated,s<sub>x</sub>=<span class=\"math-tex\">\\(\\sqrt{\\frac{f(m-\\bar{x})^2}{n-1}}\\)</span>&nbsp;where&nbsp;<em>s<sub>x</sub></em>&nbsp;= sample standard deviation&nbsp;<span class=\"math-tex\">\\(\\bar{x}=\\)</span>&nbsp;</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">sample mean; the calculations are tedious. It is usually best to use technology when performing the calculations.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">As explained before, a <em>z</em>-score allows us to compare statistics from different data sets. If the data sets have different means and standard deviations, then comparing the data values directly can be misleading.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&bull; For each data value, calculate how many standard deviations away from its mean the value is.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&bull; In symbols, the formulas for calculating <em>z</em>-scores become the following.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<table border=\"1\" cellpadding=\"1\" cellspacing=\"1\" style=\"width:900px\">\n	<tbody>\n		<tr>\n			<td>\n			<div style=\"text-align:center\"><strong><span style=\"font-size:28px\"><span style=\"color:#333333\">Sample</span></span></strong></div>\n			</td>\n			<td>\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span class=\"math-tex\">\\(Z=\\frac{x-\\bar{x}}{s}\\)</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td>\n			<div style=\"text-align:center\"><strong><span style=\"font-size:28px\"><span style=\"color:#333333\">Population</span></span></strong></div>\n			</td>\n			<td>\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span class=\"math-tex\">\\(Z=\\frac{x-\\mu}{\\sigma}\\)</span></span></div>\n			</td>\n		</tr>\n	</tbody>\n</table>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#333333\">Table 2.36</span></span></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">As shown in the table, when only a sample mean and sample standard deviation are given, the top formula is used. When the population mean and population standard deviation are given, the bottom formula is used.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:white\">Example 2.36</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Two students, John and Ali, from different high schools, wanted to find out who had the highest GPA when compared to his school. Which student had the highest GPA when compared to his school?</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong><span style=\"color:black\">Student GPA School Mean GPA School Standard Deviation</span></strong></span></p>\n\n<table cellspacing=\"0\" class=\"Table\" style=\"background:#ededed; border-collapse:collapse; border:none; width:900px\">\n	<thead>\n		<tr>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; height:15px; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong><span style=\"color:#424242\">Student</span></strong></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:15px; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong><span style=\"color:#424242\">GPA</span></strong></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:15px; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong><span style=\"color:#424242\">School Mean GPA</span></strong></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:15px; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong><span style=\"color:#424242\">School Standard Deviation</span></strong></span></div>\n			</td>\n		</tr>\n	</thead>\n	<tbody>\n		<tr>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">John</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">2.85</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">3.0</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">.7</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">Ali</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">77</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">80</span></span></div>\n			</td>\n			<td style=\"background-color:#ededed; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:16px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:#424242\">10</span></span></div>\n			</td>\n		</tr>\n	</tbody>\n</table>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Table 2.37</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution 2.36</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">For each student, determine how many standard deviations (#ofSTDEVs) his GPA is away from the average, for his school. Pay careful attention to signs when comparing and interpreting the answer.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">z </span></em><span style=\"color:black\">= # of STDEVs</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span class=\"math-tex\">\\(=\\frac{value-mean}{standard\\hspace{0.1cm}deviation}=\\frac{x+\\mu}{\\sigma}\\)</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">For John, </span><em><span style=\"color:black\">z </span></em><span style=\"color:black\">= # </span><em><span style=\"color:black\">o f STDEVs </span></em></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</span></em><span style=\"color:black\">= (2.85 &ndash; 3.0)/.07</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</span></span><span style=\"font-size:14px\"><span style=\"color:black\">= &minus; 0.21</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">For Ali, &nbsp;&nbsp;</span><em><span style=\"color:black\">z </span></em><span style=\"color:black\">= # </span><em><span style=\"color:black\">o f STDEVs</span></em></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;&nbsp; &nbsp;</span></em><span style=\"color:black\">= (77 &ndash; 80)/10</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;=&minus; 0.3</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">John has the better GPA when compared to his school because his GPA is 0.21 standard deviations </span><strong><span style=\"color:black\">below </span></strong><span style=\"color:black\">his school&#39;s mean, while Ali&#39;s GPA is .3 standard deviations </span><strong><span style=\"color:black\">below </span></strong><span style=\"color:black\">his school&#39;s mean.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">John&#39;s </span><em><span style=\"color:black\">z</span></em><span style=\"color:black\">-score of &ndash;.21 is higher than Ali&#39;s </span><em><span style=\"color:black\">z</span></em><span style=\"color:black\">-score of &ndash;.3. For GPA, higher values are better, so we conclude that John has the better GPA when compared to his school. The </span><em><span style=\"color:black\">z</span></em><span style=\"color:black\">-score representing John&#39;s score does not fall as far below the mean as the </span><em><span style=\"color:black\">z</span></em><span style=\"color:black\">-score representing Ali&#39;s score.</span></span></p>','','1','2.7.3',NULL,'1','2','1','0','1','2021-06-23 05:24:44','2021-06-23 10:54:43'),(2236,97,'The median is one of the ways to describe the data.','','multi','single','','','1','2',NULL,'1','1','1','0','1','2021-06-16 16:50:38','2021-06-16 16:50:38'),(2237,97,'This area of statistics in which you study numerical and graphical ways to describe and display your data is called Descriptive Statistics.','','multi','single','','','1','2',NULL,'1','1','1','0','1','2021-06-16 16:51:13','2021-06-16 16:51:13'),(2238,97,'When the data sets is small we use simple Graph called stem plots.','','multi','single','','','1','2.1',NULL,'1','2','1','0','1','2021-06-16 16:51:54','2021-06-16 16:51:54'),(2239,97,'The stem in stemplot graph consists of final significant digit.','','multi','single','','','1','2.1',NULL,'2','1','1','0','1','2021-06-16 16:52:28','2021-06-16 16:52:28'),(2240,97,'The number 4268 the stem __________ and leaf _______.','','multi','single','','','1','2.1',NULL,'3','1','1','0','1','2021-06-16 16:53:48','2021-06-16 16:53:48'),(2241,97,'What is the term used for an observation of data that does not fit the rest of the data.','','multi','single','','','1','2.1',NULL,'3','1','1','0','1','2021-06-16 16:55:35','2021-06-16 16:55:35'),(2242,97,'A bar graph is a straight line show the data.','','multi','single','','','1','2.1.3',NULL,'2','1','1','0','1','2021-06-16 16:56:36','2021-06-16 16:56:36'),(2243,97,'In histogram the vertical axis is labeled as','','multi','single','','','1','2.2',NULL,'4','1','1','0','1','2021-06-16 16:58:00','2021-06-16 16:58:00'),(2244,97,'The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample.','','multi','single','','','1','2.2',NULL,'1','1','1','0','1','2021-06-16 16:58:46','2021-06-16 16:58:46'),(2245,97,'A histogram is a graphic version of a frequency distribution.','','multi','single','','','1','2.2',NULL,'1','1','1','0','1','2021-06-21 11:04:50','2021-06-21 16:34:50'),(2246,97,'In a histogram the bar or interval is called','','multi','single','','','1','2.2',NULL,'1','1','1','0','1','2021-06-16 17:00:22','2021-06-16 17:00:22'),(2247,97,'Time series graph are rarely used in application of statics.','','multi','single','','','1','2.2.4',NULL,'2','1','1','0','1','2021-06-21 11:22:43','2021-06-21 16:52:43'),(2248,97,'Quartiles are not special percentiles','','multi','single','','','1','2.3',NULL,'2','1','1','0','1','2021-06-21 11:26:43','2021-06-21 16:56:43'),(2249,97,'The third quartile is the same as the 50th percentile.','','multi','single','','','1','2.3',NULL,'2','1','1','0','1','2021-06-21 11:25:34','2021-06-21 16:55:34'),(2250,97,'The Second quartile is the median.','','multi','single','','','1','2.3',NULL,'1','1','1','0','1','2021-06-16 17:03:10','2021-06-16 17:03:10'),(2251,97,'To calculate quartiles and percentiles it is necessary to order the data from smallest to largest.','','multi','single','','','1','2.3',NULL,'1','1','1','0','1','2021-06-16 17:03:55','2021-06-16 17:03:55'),(2252,97,'The median is a number that measures the ________ of the data','','multi','single','','','1','2.3',NULL,'3','1','1','0','1','2021-06-21 11:25:53','2021-06-21 16:55:53'),(2253,97,'A potential outlier is a data point that is significantly different from the other data points.','','multi','single','','','1','2.3',NULL,'1','1','1','0','1','2021-06-21 11:24:53','2021-06-21 16:54:53'),(2254,97,'High percentiles always correspond to __________ data values','','multi','single','','','1','2.3.2',NULL,'2','1','1','0','1','2021-06-22 06:02:36','2021-06-22 11:32:36'),(2255,97,'A box plot may not be useful without a scaled number line.','','multi','single','','','1','2.4',NULL,'1','1','1','0','1','2021-06-22 06:09:04','2021-06-22 11:39:04'),(2256,97,'Which word is used interchangeable for average','','multi','single','','','1','2.5',NULL,'3','1','1','0','1','2021-06-16 17:25:07','2021-06-16 17:25:07'),(2257,97,'The Greek letter ? (pronounced \"mew\") represents the','','multi','single','','','1','2.5',NULL,'2','1','1','0','1','2021-06-22 06:20:31','2021-06-22 11:50:31'),(2258,97,'The most frequent value in a data is called_______.','','multi','single','','','1','2.5',NULL,'3','1','1','0','1','2021-06-22 06:20:08','2021-06-22 11:50:08'),(2259,97,'Variability is the term used to describe the differences that may occur in these outcomes.','','multi','single','','','1','2.7.1',NULL,'1','1','1','0','1','2021-06-23 05:03:11','2021-06-23 10:33:11'),(2260,97,'3 PROBABILITY TOPICS INTRODUCTION','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">It is often necessary to </span><em><span style=\"color:black\">guess </span></em><span style=\"color:black\">about the outcome of an event in order to make a decision. Politicians study polls to guess their likelihood of winning an election. Teachers choose a particular course of study based on what they think students can comprehend. Doctors choose the treatments needed for various diseases based on their assessment of likely results. You may have visited a casino where people play games chosen because of the belief that the likelihood of winning is good. You may have chosen your course of study based on the probable availability of jobs.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">You have, more than likely, used probability. In fact, you probably have an intuitive sense of probability. Probability deals with the chance of an event occurring. Whenever you weigh the odds of whether or not to do your homework or to study for an exam, you are using probability. In this chapter, you will learn how to solve probability problems using a systematic approach.</span></span></p>','','1','3',NULL,'1','2','1','0','1','2021-06-23 05:25:53','2021-06-23 10:55:53'),(2261,97,'3.1 Terminology','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\">If the activity is a planned operation carried out under controlled conditions, it is called an <strong>experiment</strong>. If the result is <em>not </em>predetermined, then the experiment is said to be a chance experiment. Each time the experiment is attempted is called a trial.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Examples of chance experiments include the following:</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&bull; flipping a fair coin,</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&bull; spinning a spinner,</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&bull; drawing a marble at random from a bag, and </span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&bull; rolling a pair of dice.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">A result of an experiment is called an <strong>outcome</strong>. The <strong>sample space </strong>of an experiment is the set, or collection, of all possible outcomes.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<table cellspacing=\"0\" class=\"Table\" style=\"background:white; border-collapse:collapse; border:none; width:800px\">\n	<caption>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>There are four main ways to represent a sample space:</strong></span></p>\n\n	<p style=\"text-align:justify\">&nbsp;</p>\n	</caption>\n	<thead>\n		<tr>\n			<td style=\"background-color:white; border-color:black; border-style:solid; border-width:1px; height:31px; text-align:center; vertical-align:bottom\">\n			<div>&nbsp;</div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:31px; vertical-align:bottom\">\n			<div style=\"text-align:center\"><br />\n			<span style=\"font-size:14px\"><span style=\"color:black\">Flipping a Fair Coin</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:31px; vertical-align:top; width:179px\">\n			<div style=\"text-align:center\">&nbsp;</div>\n\n			<div style=\"text-align:center\">&nbsp;</div>\n\n			<div style=\"text-align:center\"><span style=\"font-size:14px\">Flipping Two Fair Coins</span></div>\n			</td>\n		</tr>\n	</thead>\n	<tbody>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:67px\">\n			<div style=\"text-align:center\"><span style=\"font-size:14px\"><strong><span style=\"color:black\">Systematic List of Outcomes</span></strong></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:67px\">\n			<div style=\"text-align:center\"><span style=\"font-size:14px\"><span style=\"color:#424242\">heads (H)<br />\n			tails (T)</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:67px\">\n			<div style=\"text-align:center\"><span style=\"font-size:14px\"><span style=\"color:#424242\">HH<br />\n			HT<br />\n			TH<br />\n			TT</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:67px; width:216px\">\n			<div style=\"text-align:center\"><span style=\"font-size:14px\"><strong><span style=\"color:black\">Tree Diagram*</span></strong></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:67px; width:244px\">\n			<div style=\"text-align:center\"><span style=\"font-size:14px\"><span style=\"color:black\"><img alt=\"https://openstax.org/apps/archive/20210421.141058/resources/338af2eaeec383a3051027bf9449d7448ee8ec79\" id=\"_x0031_\" src=\"https://openstax.org/apps/archive/20210421.141058/resources/338af2eaeec383a3051027bf9449d7448ee8ec79\" style=\"height:33.75pt; width:75pt\" /> </span></span></div>\n\n			<div style=\"text-align:center\"><span style=\"font-size:14px\"><span style=\"color:black\">Table3.1&nbsp;Figure3.2</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:67px\">\n			<div style=\"text-align:center\"><span style=\"font-size:14px\"><span style=\"color:black\"><img alt=\"https://openstax.org/apps/archive/20210421.141058/resources/e565bab541fe9493628b8cbe535cd0292a8740bd\" id=\"_x0032_\" src=\"https://openstax.org/apps/archive/20210421.141058/resources/e565bab541fe9493628b8cbe535cd0292a8740bd\" style=\"height:45.75pt; width:75pt\" /> </span></span></div>\n\n			<div style=\"text-align:center\"><span style=\"font-size:14px\"><span style=\"color:black\">Figure3.3</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:67px\">\n			<div style=\"text-align:center\"><span style=\"font-size:14px\"><strong><span style=\"color:black\">Venn Diagram*</span></strong></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:67px\">\n			<div style=\"text-align:center\"><span style=\"font-size:14px\"><span style=\"color:black\"><img alt=\"https://openstax.org/apps/archive/20210421.141058/resources/e6a1f399f79be6f5169e2b98d5cf0d989da815bd\" id=\"_x0033_\" src=\"https://openstax.org/apps/archive/20210421.141058/resources/e6a1f399f79be6f5169e2b98d5cf0d989da815bd\" style=\"height:72.75pt; width:75pt\" /> </span></span></div>\n\n			<div style=\"text-align:center\"><span style=\"font-size:14px\"><span style=\"color:black\">Figure3.4</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:67px\">\n			<div style=\"text-align:center\"><span style=\"font-size:14px\"><span style=\"color:black\"><img alt=\"https://openstax.org/apps/archive/20210421.141058/resources/25c26adee9586c0242e7f1af91a2700b84f7b442\" id=\"_x0034_\" src=\"https://openstax.org/apps/archive/20210421.141058/resources/25c26adee9586c0242e7f1af91a2700b84f7b442\" style=\"height:58.5pt; width:95.25pt\" /> </span></span></div>\n\n			<div style=\"text-align:center\"><span style=\"font-size:14px\"><span style=\"color:black\">Figure3.5</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:67px\">\n			<div style=\"text-align:center\"><span style=\"font-size:14px\"><strong><span style=\"color:black\">Set Notation</span></strong></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:67px\">\n			<div style=\"text-align:center\"><span style=\"font-size:14px\"><span style=\"color:black\">S=&nbsp;{H,&nbsp;T}</span></span></div>\n			</td>\n			<td style=\"background-color:white; border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:67px\">\n			<div style=\"text-align:center\"><span style=\"font-size:14px\"><span style=\"color:black\">S=&nbsp;{HH,&nbsp;HT,&nbsp;TH,&nbsp;TT}</span></span></div>\n			</td>\n		</tr>\n	</tbody>\n</table>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">We will investigate tree diagrams and Venn diagrams in Section 3.5.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">Note&mdash;when represented as a set, the sample space is denoted with an uppercase&nbsp;<em>S</em>.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">An event is any combination of outcomes. It is a subset of the sample space, so uppercase letters like&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;are commonly used to represent events. For example, if the experiment is to flip three fair coins, event&nbsp;<em>A</em>&nbsp;might be getting at most one head.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The probability of an event&nbsp;<em>A</em>&nbsp;is written&nbsp;<em>P</em>(<em>A</em>), and&nbsp;0<span style=\"background-color:white\"><span style=\"color:black\">&le;&nbsp;P(A)&nbsp;&le;&nbsp;1.P(A)&nbsp;=&nbsp;0</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">means the event&nbsp;<em>A</em>&nbsp;can never happen.&nbsp;<em>P</em>(<em>A</em>) = 1 means the event&nbsp;<em>A</em>&nbsp;always happens.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">P(A)&nbsp;=&nbsp;0.5 means the event&nbsp;<em>A</em>&nbsp;is&nbsp;<strong>equally likely</strong>&nbsp;to occur or not to occur.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238475841623847584.jpg\" style=\"height:121px; width:400px\" /></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\">Figure 3.6</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">If two outcomes or events are equally likely, then they have equal probability. For example, if you toss a fair, six-sided die, each face (1, 2, 3, 4, 5, or 6) is as likely to occur as any other face. If you toss a fair coin, a Head (<em>H</em>) and a Tail (<em>T</em>) are equally likely to occur. If you randomly guess the answer to a true/false question on an exam, you are equally likely to select a correct answer or an incorrect answer.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">To calculate the probability of an event&nbsp;<em>A</em>&nbsp;when all outcomes in the sample space are equally likely, count the number of outcomes for event&nbsp;<em>A</em>&nbsp;and divide by the total number of outcomes in the sample space. This is known as the theoretical probability of&nbsp;<em>A</em>.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>Theoretical Probability of Event A</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span class=\"math-tex\">\\(P(A)\\)</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span class=\"math-tex\">\\(=\\frac{Number\\hspace{0.1cm} of\\hspace{0.1cm}outcomes\\hspace{0.1cm}in\\hspace{0.1cm}event }{A\\hspace{0.1cm}total\\hspace{0.1cm}number\\hspace{0.1cm}of\\hspace{0.1cm}possible\\hspace{0.1cm}outcomes}\\)</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">For example, if you toss a fair dime and a fair nickel, the sample space is {<em>HH, TH, HT, TT</em>} where&nbsp;<em>T</em>&nbsp;= tails and&nbsp;<em>H</em>&nbsp;= heads. The sample space has four outcomes. Let&nbsp;<em>A</em>&nbsp;represent the outcome&nbsp;<em>getting one head.</em>&nbsp;There are two outcomes that meet this condition {<em>HT, TH</em>}, so</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">P(A)=24=12=.5.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">Theoretical probability is not sufficient in all situations, however. Suppose we want to calculate the probability that a randomly selected car will run a red light at a given intersection. In this case, we need to look at events that&nbsp;<em>have</em>&nbsp;occurred, not theoretical possibilities. We could install a traffic camera and count the number of times that cars failed to stop when the light was red and the total number of cars that passed through the intersection for a period of time. These data will allow us to calculate the experimental, or empirical, probability that a car runs the red light.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>Experimental Probability of Event A</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span class=\"math-tex\">\\(P(A)\\)</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span class=\"math-tex\">\\(=\\frac{Number\\hspace{0.1cm} of\\hspace{0.1cm}times\\hspace{0.1cm}event\\hspace{0.1cm}A\\hspace{0.1cm}occurs }{Total\\hspace{0.1cm}number\\hspace{0.1cm}of\\hspace{0.1cm}trials}\\)</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">While theoretical and experimental methods provide two different ways to calculate probability, these methods are closely related. If you flip one fair coin, there is one way to obtain heads and two possible outcomes. So, the theoretical probability of heads is&nbsp;</span><span class=\"math-tex\">\\(\\frac{1}{2}\\)</span>.&nbsp;<span style=\"background-color:white\"><span style=\"color:black\">Probability does not predict short-term results, however. If an experiment involves flipping a coin 10 times, you should not expect exactly five heads and five tails. The probability of any outcome measures the long-term relative frequency of that outcome. If you continue to flip the coin (from 20 to 2,000 to 20,000 times) the relative frequency of heads approaches .5 (the probability of heads).This important characteristic of probability experiments is known as the law of large numbers, which states that as the number of repetitions of an experiment is increased, the relative frequency obtained in the experiment tends to become closer and closer to the theoretical probability. Even though the outcomes do not happen according to any set pattern or order, overall, the long-term observed, or empirical, relative frequency will approach the theoretical probability.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">Suppose you roll one fair, six-sided die with the numbers {1, 2, 3, 4, 5, 6} on its faces. Let event&nbsp;<em>E</em>&nbsp;= rolling a number that is at least five. There are two outcomes {5, 6}.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\"><span class=\"math-tex\">\\(P(E)=\\frac{2}{6}\\)</span></span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">If you were to roll the die only a few times, you would not be surprised if your observed results did not match the probability. If you were to roll the die a very large number of times, you would expect that, overall,&nbsp;<span class=\"math-tex\">\\(\\frac{2}{6}\\)</span>&nbsp;of the rolls would result in an outcome of&nbsp;<em>at least five</em>. You would not expect exactly&nbsp;<span class=\"math-tex\">\\(\\frac{2}{6}\\)</span>,but the long-term relative frequency of obtaining this result would approach the theoretical probability of&nbsp;<span class=\"math-tex\">\\(\\frac{2}{6}\\)</span></span><span style=\"background-color:white\"><span style=\"color:black\">as the number of repetitions grows larger and larger.</span></span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">It is important to realize that in many situations, the outcomes are not equally likely. A coin or die may be&nbsp;<strong>unfair</strong>, or&nbsp;<strong>biased</strong>. Two math professors in Europe had their statistics students test the Belgian one-euro coin and discovered that in 250 trials, a head was obtained 56 percent of the time and a tail was obtained 44 percent of the time. The data seem to show that the coin is not a fair coin; more repetitions would be helpful to draw a more accurate conclusion about such bias. Some dice may be biased. Look at the dice in a game you have at home; the spots on each face are usually small holes carved out and then painted to make the spots visible. Your dice may or may not be biased; it is possible that the outcomes may be affected by the slight weight differences due to the different numbers of holes in the faces. Gambling casinos make a lot of money depending on outcomes from rolling dice, so casino dice are made differently to eliminate bias. Casino dice have&nbsp;</span><span style=\"background-color:white\"><span style=\"color:black\">flat faces; the holes are completely filled with paint having the same density as the material that the dice are made out of so that each face is equally likely to occur. Later we will learn techniques to use to work with probabilities for events that are not equally likely.</span></span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong><span style=\"background-color:white\"><span style=\"color:black\">OR Event </span></span></strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"background-color:white\"><span style=\"color:black\">An outcome is in the event&nbsp;<em>A</em>&nbsp;OR&nbsp;<em>B</em>&nbsp;if the outcome is in&nbsp;<em>A</em>&nbsp;or is in&nbsp;<em>B</em>&nbsp;or is in both&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>. For example, let&nbsp;<em>A</em>&nbsp;= {1, 2, 3, 4, 5} and&nbsp;<em>B</em>&nbsp;= {4, 5, 6, 7, 8}.&nbsp;<em>A</em>&nbsp;OR&nbsp;<em>B</em>&nbsp;= {1, 2, 3, 4, 5, 6, 7, 8}. Notice that 4 and 5 are&nbsp;<strong>not</strong>&nbsp;listed twice.</span></span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>AND Event</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">An outcome is in the event&nbsp;<em>A</em>&nbsp;AND&nbsp;<em>B</em>&nbsp;if the outcome is in both&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;at the same time. For example, let&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;be {1, 2, 3, 4, 5} and {4, 5, 6, 7, 8}, respectively. Then&nbsp;<em>A</em>&nbsp;AND&nbsp;<em>B</em>&nbsp;= {4, 5}.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">The&nbsp;<strong>complement</strong>&nbsp;of event&nbsp;<em>A</em>&nbsp;is denoted&nbsp;<em>A&prime;</em>&nbsp;(read &quot;<em>A</em>&nbsp;prime&quot;).&nbsp;<em>A&prime;</em>&nbsp;consists of all outcomes that are&nbsp;<strong>not</strong>&nbsp;in&nbsp;<em>A</em>. Notice that&nbsp;</span></span><em><span style=\"background-color:white\"><span style=\"color:black\">P</span></span></em><span style=\"background-color:white\"><span style=\"color:black\">(<em>A</em>) +&nbsp;<em>P</em>(<em>A&prime;</em>) = 1. For example, let&nbsp;<em>S</em>&nbsp;= {1, 2, 3, 4, 5, 6} and let&nbsp;<em>A</em>&nbsp;= {1, 2, 3, 4}. Then,&nbsp;<em>A&prime;</em>&nbsp;= {5, 6}.&nbsp;<em>P</em>(<em>A</em>) =&nbsp;4646,&nbsp;<em>P</em>(<em>A&prime;</em>) =&nbsp;2626, and&nbsp;<em>P</em>(<em>A</em>) +&nbsp;<em>P</em>(<em>A&prime;</em>) =&nbsp;46+2646&nbsp;+&nbsp;26&nbsp;= 1.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The&nbsp;<strong>conditional probability</strong>&nbsp;of&nbsp;<em>A</em>&nbsp;given&nbsp;<em>B</em>&nbsp;is written&nbsp;<em>P</em>(<em>A</em>|<em>B</em>), read &quot;the probability of&nbsp;<em>A</em>, given&nbsp;<em>B</em>.&quot;&nbsp;<em>P</em>(<em>A</em>|<em>B</em>) is the probability that event&nbsp;<em>A</em>&nbsp;will occur given that the event&nbsp;<em>B</em>&nbsp;has already occurred.&nbsp;<strong>A conditional probability reduces the sample space</strong>. We calculate the probability of&nbsp;<em>A</em>&nbsp;from the reduced sample space&nbsp;<em>B</em>. The formula to calculate&nbsp;<em>P</em>(<em>A</em>|<em>B</em>) is&nbsp;<em>P</em>(<em>A</em>|<em>B</em>) <span class=\"math-tex\">\\(=\\frac{P(A\\hspace{0.1cm}AND\\hspace{0.1cm}B)}{P(B)}\\)</span>&nbsp;where&nbsp;<em>P</em>(<em>B</em>) is greater than zero.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">For example, suppose we toss one fair, six-sided die. The sample space&nbsp;<em>S</em>&nbsp;= {1, 2, 3, 4, 5, 6}. Let&nbsp;<em>A</em>&nbsp;= {2, 3} and&nbsp;<em>B</em>&nbsp;= {2, 4, 6}.&nbsp;<em>P</em>(<em>A</em>|<em>B</em>) represents the probability that a randomly selected outcome is in&nbsp;<em>A given that</em>&nbsp;it is in&nbsp;<em>B</em>. We know that the outcome must lie in&nbsp;<em>B</em>, so there are three possible outcomes. There is only one outcome in&nbsp;<em>B</em>&nbsp;that also lies in&nbsp;<em>A</em>, so&nbsp;<em>P</em>(<em>A</em>|<em>B</em>) <span class=\"math-tex\">\\(=\\frac{1}{3}\\)</span>.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">We get the same result by using the formula. Remember that&nbsp;<em>S</em>&nbsp;has six outcomes.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span class=\"math-tex\">\\(P(\\frac{A}{B})=\\frac{P(A\\hspace{0.1cm}AND\\hspace{0.1cm}B)}{P(B)}\\)</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span class=\"math-tex\">\\(=\\frac{\\frac{(the\\hspace{0.1cm}number\\hspace{0.1cm}of\\hspace{0.1cm}outcomes\\hspace{0.1cm}that\\hspace{0.1cm}are\\hspace{0.1cm}2\\hspace{0.1cm}or\\hspace{0.1cm}3\\hspace{0.1cm}and\\hspace{0.1cm}even\\hspace{0.1cm}in\\hspace{0.1cm}S)}{6}}{\\frac{(the\\hspace{0.1cm}number\\hspace{0.1cm}of\\hspace{0.1cm}outcomes\\hspace{0.1cm}that\\hspace{0.1cm}are\\hspace{0.1cm}even\\hspace{0.1cm}in\\hspace{0.1cm}S)}{6}}\\)</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&nbsp;&nbsp;<span class=\"math-tex\">\\(=\\frac{\\frac{1}{6}}{\\frac{3}{6}}=\\frac{1}{3}\\)</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>Understanding Terminology and Symbols</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">It is important to read each problem carefully to think about and understand what the events are. Understanding the wording is the first very important step in solving probability problems. Reread the problem several times if necessary. Clearly identify the event of interest. Determine whether there is a condition stated in the wording that would indicate that the probability is conditional; carefully identify the condition, if any.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&nbsp;</span></p>','','1','3.1',NULL,'1','2','1','0','1','2021-06-23 05:47:19','2021-06-23 11:17:19'),(2262,97,'3.1.1 EXAMPLES','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>EXAMPLE&nbsp;3.1</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The sample space&nbsp;<em>S</em>&nbsp;is the whole numbers starting at one and less than 20.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<ol style=\"list-style-type:lower-alpha\">\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>S</em>&nbsp;= ________&nbsp;</span></p>\n	</li>\n</ol>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Let event&nbsp;<em>A</em>&nbsp;= the even numbers and event&nbsp;<em>B</em>&nbsp;= numbers greater than 13.</span></p>\n\n<p style=\"margin-left:48px; text-align:justify\">&nbsp;</p>\n\n<ol start=\"2\" style=\"list-style-type:lower-alpha\">\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>A</em>&nbsp;= ________,&nbsp;<em>B</em>&nbsp;= ________</span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>P</em>(<em>A</em>) = ________,&nbsp;<em>P</em>(<em>B</em>) = ________</span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>A</em>&nbsp;AND&nbsp;<em>B</em>&nbsp;= ________,&nbsp;<em>A</em>&nbsp;OR&nbsp;<em>B</em>&nbsp;= ________</span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) = ________,&nbsp;<em>P</em>(<em>A</em>&nbsp;OR&nbsp;<em>B</em>) = ________</span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>A&prime;</em>&nbsp;= ________,&nbsp;<em>P</em>(<em>A&prime;</em>) = ________</span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>P</em>(<em>A</em>) +&nbsp;<em>P</em>(<em>A&prime;</em>) = ________</span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em>P</em>(<em>A</em>|<em>B</em>) = ________,&nbsp;<em>P</em>(<em>B</em>|<em>A</em>) = ________; are the probabilities</span></p>\n	</li>\n</ol>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong><span style=\"color:black\">Solution&nbsp;3.1</span></strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<ol style=\"list-style-type:lower-alpha\">\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">S</span></em><span style=\"color:black\">&nbsp;= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">A</span></em><span style=\"color:black\">&nbsp;= {2, 4, 6, 8, 10, 12, 14, 16, 18},&nbsp;<em>B</em>&nbsp;= {14, 15, 16, 17, 18, 19}</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span class=\"math-tex\">\\(P(A)=\\frac{Number\\hspace{0.1cm} of\\hspace{0.1cm}outcomes\\hspace{0.1cm}in\\hspace{0.1cm}A }{Number\\hspace{0.1cm}of\\hspace{0.1cm}outcomesin\\hspace{0.1cm}s}=\\frac{1}{19}\\)</span></span><span style=\"font-size:14px\">&nbsp;&nbsp;<span class=\"math-tex\">\\(P(A)=\\frac{Number\\hspace{0.1cm} of\\hspace{0.1cm}outcomes\\hspace{0.1cm}in\\hspace{0.1cm}B }{Number\\hspace{0.1cm}of\\hspace{0.1cm}outcomesin\\hspace{0.1cm}s}ins=\\frac{6}{19}\\)</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The set&nbsp;<em>A</em>&nbsp;AND&nbsp;<em>B</em>&nbsp;contains all outcomes that lie in both sets&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>, so&nbsp;<em>A</em>&nbsp;AND&nbsp;<em>B</em>&nbsp;= {14,16,18}, The set&nbsp;<em>A</em>&nbsp;OR&nbsp;<em>B</em>&nbsp;contains all outcomes that lie either of the sets&nbsp;<em>A</em>&nbsp;or&nbsp;<em>B</em>, so&nbsp;<em>A</em>&nbsp;OR&nbsp;<em>B</em>&nbsp;= {2, 4, 6, 8, 10, 12, 14, 15, 16, 17, 18, 19}.</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) =&nbsp;</span><span class=\"math-tex\">\\(\\frac{3}{19}\\)</span>&nbsp;<span style=\"color:black\">,&nbsp;<em>P</em>(<em>A</em>&nbsp;OR&nbsp;<em>B</em>) =&nbsp;</span><span class=\"math-tex\">\\(\\frac{12}{19}\\)</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">A&#39;</span></em><span style=\"color:black\">&nbsp;consists of all outcomes in the sample space,&nbsp;<em>S</em>, that DO NOT lie in&nbsp;<em>A</em>, so&nbsp;<em>A&prime;</em>&nbsp;= 1, 3, 5, 7, 9, 11, 13, 15, 17, 19;&nbsp;<em>P</em>(<em>A&prime;</em>) =&nbsp;</span><span class=\"math-tex\">\\(\\frac{12}{19}\\)</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\"><em>&nbsp;P</em>(<em>A</em>) +&nbsp;<em>P</em>(<em>A&prime;</em>) =<span class=\"math-tex\">\\(\\frac{9}{19}+\\frac{10}{19}=1\\)</span></span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\"><em>P</em>(<em>A</em>|<em>B</em>) &nbsp;</span><strong><span style=\"color:black\"><span class=\"math-tex\">\\(=\\frac{P(A\\hspace{0.1cm}AND\\hspace{0.1cm}B)}{P(B)}\\)</span></span></strong></span></p>\n	</li>\n</ol>\n\n<p style=\"margin-left:24px; text-align:justify\"><span style=\"font-size:14px\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<span class=\"math-tex\">\\(=\\frac{\\frac{3}{19}}{\\frac{6}{19}}\\)</span></span></p>\n\n<p style=\"margin-left:24px; text-align:justify\"><span style=\"font-size:14px\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"math-tex\">\\(=\\frac{3}{6}P(BAB)\\)</span></span></p>\n\n<p style=\"margin-left:24px; text-align:justify\"><span style=\"font-size:14px\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<span class=\"math-tex\">\\(=\\frac{P(A\\hspace{0.1cm}AND\\hspace{0.1cm}B)}{P(A)}\\)</span></span></p>\n\n<p style=\"margin-left:24px; text-align:justify\"><span style=\"font-size:14px\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<span class=\"math-tex\">\\(=\\frac{\\frac{3}{19}}{\\frac{9}{19}}\\)</span></span></p>\n\n<p style=\"margin-left:24px; text-align:justify\"><span style=\"font-size:14px\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class=\"math-tex\">\\(=\\frac{3}{9}\\)</span></span></p>\n\n<p style=\"margin-left:24px; text-align:justify\">&nbsp;</p>\n\n<p style=\"margin-left:24px; text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">No, the probabilities are not equal.&nbsp;</span></span></p>\n\n<p style=\"margin-left:24px; text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&nbsp; </span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>EXAMPLE&nbsp;3.2</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">A fair, six-sided die is rolled. The sample space,&nbsp;<em>S</em>, is {1, 2, 3, 4, 5, 6}. Describe each event and calculate its probability.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<ol style=\"list-style-type:lower-alpha\">\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Event&nbsp;<em>T</em>&nbsp;= the outcome is two.</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Event&nbsp;<em>A</em>&nbsp;= the outcome is an even number.</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Event&nbsp;<em>B</em>&nbsp;= the outcome is less than four.</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The complement of&nbsp;<em>A</em></span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">A</span></em><span style=\"color:black\">&nbsp;GIVEN&nbsp;<em>B</em></span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">B</span></em><span style=\"color:black\">&nbsp;GIVEN&nbsp;<em>A</em></span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">A</span></em><span style=\"color:black\">&nbsp;AND&nbsp;<em>B</em></span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">A</span></em><span style=\"color:black\">&nbsp;OR&nbsp;<em>B</em></span></span></p>\n	</li>\n	<li value=\"1\">\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">A</span></em><span style=\"color:black\">&nbsp;OR&nbsp;<em>B&prime;</em></span></span></p>\n	</li>\n	<li value=\"10\">\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Event&nbsp;<em>N</em>&nbsp;= the outcome is a prime number.</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Event&nbsp;<em>I</em>&nbsp;= the outcome is seven.</span></span></p>\n	</li>\n</ol>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>Solution&nbsp;3.2</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<ol start=\"1\" style=\"list-style-type:lower-alpha\">\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">T</span></em><span style=\"color:black\">&nbsp;= {2},&nbsp;<em>P</em>(<em>T</em>)</span><span style=\"color:black\"> </span><span class=\"math-tex\">\\(=\\frac{NUMBER\\hspace{0.1cm}OF\\hspace{0.1cm}OUTCOMES\\hspace{0.1cm}IN\\hspace{0.1cm}T}{NUMBER\\hspace{0.1cm}OF\\hspace{0.1cm}OUTCOMES\\hspace{0.1cm}IN\\hspace{0.1cm}S}=\\frac{1}{6}\\)</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">A</span></em><span style=\"color:black\">&nbsp;= {2, 4, 6},&nbsp;<em>P</em>(<em>A</em>) <span class=\"math-tex\">\\(=\\frac{3}{6}=\\frac{1}{2}\\)</span></span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">B</span></em><span style=\"color:black\">&nbsp;= {1, 2, 3},&nbsp;<em>P</em>(<em>B</em>)&nbsp;<span class=\"math-tex\">\\(=\\frac{3}{6}=\\frac{1}{2}\\)</span></span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">A&prime;</span></em><span style=\"color:black\">&nbsp;= {1, 3, 5},&nbsp;<em>P</em>(<em>A&prime;</em>)&nbsp;<span class=\"math-tex\">\\(=\\frac{3}{6}=\\frac{1}{2}\\)</span></span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">A</span></em><span style=\"color:black\">|<em>B</em>&nbsp;= {2}, There are three outcomes in&nbsp;<em>B</em>, and only 1 of these lies in&nbsp;<em>A</em>, so&nbsp;<em>P</em>(<em>A</em>|<em>B</em>) </span><span class=\"math-tex\">\\(=\\frac{1}{3}\\)</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">B</span></em><span style=\"color:black\">|<em>A</em>&nbsp;= {2}, There are three outcomes in&nbsp;<em>A</em>, and only 1 of these lies in&nbsp;<em>B</em>, so&nbsp;<em>P</em>(<em>B</em>|<em>A</em>)&nbsp;</span><span class=\"math-tex\">\\(=\\frac{1}{3}\\)</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">A</span></em><span style=\"color:black\">&nbsp;AND&nbsp;<em>B</em>&nbsp;= {2},&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>)&nbsp;</span><span class=\"math-tex\">\\(=\\frac{1}{6}\\)</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">A</span></em><span style=\"color:black\">&nbsp;OR&nbsp;<em>B</em>&nbsp;= {1, 2, 3, 4, 6},&nbsp;<em>P</em>(<em>A</em>&nbsp;OR&nbsp;<em>B</em>)&nbsp;</span><span class=\"math-tex\">\\(=\\frac{5}{6}\\)</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">A</span></em><span style=\"color:black\">&nbsp;OR&nbsp;<em>B&prime;</em>&nbsp;= {2, 4, 5, 6},&nbsp;<em>P</em>(<em>A</em>&nbsp;OR&nbsp;<em>B&prime;</em>)&nbsp;</span><span class=\"math-tex\">\\(=\\frac{4}{6}=\\frac{2}{3}\\)</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">N</span></em><span style=\"color:black\">&nbsp;= {2, 3, 5},&nbsp;<em>P</em>(<em>N</em>)&nbsp;</span><span class=\"math-tex\">\\(=\\frac{1}{2}\\)</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">It is impossible to roll a die and get an outcome of 7, so&nbsp;<em>P</em>(7) = 0.</span></span></p>\n	</li>\n</ol>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong><span style=\"color:black\">EXAMPLE&nbsp;3.3</span></strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><a href=\"https://openstax.org/books/statistics/pages/3-1-terminology#ch03_M02-tbl001\" style=\"color:blue; text-decoration:underline\"><span style=\"color:#027eb5\">Table 3.2</span></a>&nbsp;<span style=\"color:black\">describes the distribution of a random sample&nbsp;<em>S</em>&nbsp;of 100 individuals, organized by gender and whether they are right or left-handed</span><span style=\"color:#424242\">.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<table cellspacing=\"0\" class=\"Table\" style=\"border-collapse:collapse; border:none; width:900px\">\n	<thead>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\">&nbsp;</div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Right-Handed</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Left-Handed</strong></span></div>\n			</td>\n		</tr>\n	</thead>\n	<tbody>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">Males</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">43</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">9</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">Females</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">44</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">4</span></div>\n			</td>\n		</tr>\n	</tbody>\n</table>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#424242\">Let&rsquo;s denote the events&nbsp;<em>M</em>&nbsp;= the subject is male,&nbsp;<em>F</em>&nbsp;= the subject is female,&nbsp;<em>R</em>&nbsp;= the subject is right-handed,&nbsp;<em>L</em>&nbsp;= the subject is left-handed. Compute the following probabilities:</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<ol style=\"list-style-type:lower-alpha\">\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#424242\"><em>P</em>(<em>M</em>)</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#424242\"><em>P</em>(<em>F</em>)</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#424242\"><em>P</em>(<em>R</em>)</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#424242\"><em>P</em>(<em>L</em>)</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#424242\"><em>P</em>(<em>M</em>&nbsp;AND&nbsp;<em>R</em>)</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#424242\"><em>P</em>(<em>F</em>&nbsp;AND&nbsp;<em>L</em>)</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#424242\"><em>P</em>(<em>M</em>&nbsp;OR&nbsp;<em>F</em>)</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#424242\"><em>P</em>(<em>M</em>&nbsp;OR&nbsp;<em>R</em>)</span></span></p>\n	</li>\n	<li value=\"1\">\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#424242\"><em>P</em>(<em>F</em>&nbsp;OR&nbsp;<em>L</em>)</span></span></p>\n	</li>\n	<li value=\"10\">\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#424242\"><em>P</em>(<em>M&#39;</em>)</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#424242\"><em>P</em>(<em>R</em>|<em>M</em>)</span></span></p>\n	</li>\n	<li value=\"50\">\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#424242\"><em>P</em>(<em>F</em>|<em>L</em>)</span></span></p>\n	</li>\n	<li value=\"1000\">\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#424242\"><em>P</em>(<em>L</em>|<em>F</em>)</span></span></p>\n	</li>\n</ol>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#424242\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238609431623860943.png\" style=\"height:834px; width:800px\" /></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\">&nbsp;</p>','','1','3.1.1',NULL,'1','2','1','0','1','2021-06-23 06:03:38','2021-06-23 11:33:38'),(2263,97,'3.2 Independent and Mutually Exclusive Events','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Independent and mutually exclusive do </span><strong><span style=\"color:black\">not </span></strong><span style=\"color:black\">mean the same thing.</span></span></p>','','1','3.2',NULL,'1','2','1','0','1','2021-06-23 06:04:32','2021-06-23 11:34:32'),(2264,97,'3.2.1 Independent Events','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Two events are independent if the following are true:</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">A</span></em><span style=\"color:black\">|</span><em><span style=\"color:black\">B</span></em><span style=\"color:black\">) = </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">A</span></em><span style=\"color:black\">)</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">B</span></em><span style=\"color:black\">|</span><em><span style=\"color:black\">A</span></em><span style=\"color:black\">) = </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">B</span></em><span style=\"color:black\">)</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&bull; </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">A </span></em><span style=\"color:black\">AND </span><em><span style=\"color:black\">B</span></em><span style=\"color:black\">) = </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">A</span></em><span style=\"color:black\">)</span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">B</span></em><span style=\"color:black\">)</span></span></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:#424242\">Two events&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;are&nbsp;<strong>independent events</strong>&nbsp;if the knowledge that one occurred does not affect the chance the other occurs. For example, the outcomes of two roles of a fair die are independent events. The outcome of the first roll does not change the probability for the outcome of the second roll. To show two events are independent, you must show&nbsp;<strong>only one</strong>&nbsp;of the above conditions. If two events are&nbsp;<strong>not</strong>&nbsp;independent, then we say that they are&nbsp;<strong>dependent events</strong>.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:#424242\">Sampling may be done&nbsp;<strong>with</strong>&nbsp;<strong>replacement</strong>&nbsp;or&nbsp;<strong>without replacement</strong>.</span></span></span></p>\n\n<p>&nbsp;</p>\n\n<ul>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:#424242\"><strong><span style=\"color:#0070c0\">With replacement</span></strong>: If each member of a population is replaced after it is picked, then that member has the possibility of being chosen more than once. When sampling is done with replacement, then events are considered to be independent, meaning the result of the first pick will not change the probabilities for the second pick.</span></span></span></p>\n	</li>\n</ul>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:#424242\">???????</span></span><span style=\"color:#424242\">A bag contains four blue and three white marbles. James draws one marble from the bag at random, records the color, and replaces the marble. The probability of drawing blue is&nbsp;<span class=\"math-tex\">\\(\\frac{4}{7}\\)</span>. When James draws a marble from the bag a second time, the probability of drawing blue is still&nbsp;<span class=\"math-tex\">\\(\\frac{4}{7}\\)</span>.</span><span style=\"background-color:white\"><span style=\"color:#424242\">James replaced the marble after the first draw, so there are still four blue and three white marbles.</span> </span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238617671623861767.jpg\" style=\"height:501px; width:400px\" /></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong><span style=\"background-color:white\"><span style=\"color:#424242\">Figure&nbsp;3.7</span></span></strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">&nbsp;<strong><span style=\"color:#424242\">Without replacement</span></strong><span style=\"color:#424242\">: </span><span style=\"color:black\">When sampling is done without replacement, each member of a population may be chosen only once. In this case, the probabilities for the second pick are affected by the result of the first pick. The events are considered to be dependent or not independent.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The bag still contains four blue and three white marbles. Maria draws one marble from the bag at random, records the color, and sets the marble aside. The probability of drawing blue on the first draw is&nbsp;</span><span style=\"color:#424242\"><span class=\"math-tex\">\\(\\frac{4}{7}\\)</span></span><span style=\"color:black\">. Suppose Maria draws a blue marble and sets it aside. When she draws a marble from the bag a second time, there are now&nbsp;<em>three</em>&nbsp;blue and three white marbles. So, the probability of drawing blue is now&nbsp;</span><span style=\"color:#424242\"><span class=\"math-tex\">\\(\\frac{3}{6}=\\frac{1}{2}\\)</span></span><span style=\"background-color:white\"><span style=\"color:black\">. Removing the first marble without replacing it influences the probabilities on the second draw.</span> <span style=\"background-color:white\"><span style=\"color:black\">If it is not known whether&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;are independent or dependent,&nbsp;<strong>assume they are dependent until you can show otherwise</strong>.</span></span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"background-color:white\"><span style=\"color:black\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238619131623861913.png\" style=\"height:359px; width:400px\" /></span></span><span style=\"color:black\">Fig 3.8</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><strong>EXAMPLE&nbsp;3.4</strong></span></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">You have a fair, well-shuffled deck of 52 cards. It consists of four suits. The suits are clubs, diamonds, hearts, and spades. Clubs and spades are black, while diamonds and hearts are red cards. There are 13 cards in each suit consisting of&nbsp;<em>A</em>&nbsp;(ace), 2, 3, 4, 5, 6, 7, 8, 9, 10,&nbsp;<em>J</em>&nbsp;(jack),&nbsp;<em>Q</em>&nbsp;(queen),&nbsp;<em>K</em>&nbsp;(king) of that suit.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><img alt=\"\" src=\"https://edupmquestionhelp.s3.ap-south-1.amazonaws.com/hint/uploads/images/16238619731623861973.jpg\" style=\"height:363px; width:800px\" /></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>Figure&nbsp;3.9</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>Sampling with replacement</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Suppose you pick three cards with replacement. The first card you pick out of the 52 cards is the&nbsp;<em>Q</em>&nbsp;of spades. You put this card back, reshuffle the cards and pick a second card from the 52-card deck. It is the 10 of clubs. You put this card back, reshuffle the cards and pick a third card from the 52-card deck. This time, the card is the&nbsp;<em>Q</em>&nbsp;of spades again. Your picks are {<em>Q</em>&nbsp;of spades, 10 of clubs,&nbsp;<em>Q</em>&nbsp;of spades}. You have picked the&nbsp;<em>Q</em>&nbsp;of spades twice. You pick each card from the 52-card deck.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><strong>Sampling without replacement</strong></span></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Suppose you pick three cards without replacement. The first card you pick out of the 52 cards is the&nbsp;<em>K</em>&nbsp;of hearts. You put this card aside and pick the second card from the 51 cards remaining in the deck. It is the three of diamonds. You put this card aside and pick the third card from the remaining 50 cards in the deck. The third card is the&nbsp;<em>J</em>&nbsp;of spades. Your picks are {<em>K</em>&nbsp;of hearts, three of diamonds,&nbsp;<em>J</em>&nbsp;of spades}. Because you have picked the cards without replacement, you cannot pick the same card twice.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>EXAMPLE&nbsp;3.5</strong></span></span></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">You have a fair, well-shuffled deck of 52 cards. It consists of four suits. The suits are clubs, diamonds, hearts, and spades. There are 13 cards in each suit consisting of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,&nbsp;<em>J</em>&nbsp;(jack),&nbsp;<em>Q</em>&nbsp;(queen), and&nbsp;<em>K</em>&nbsp;(king) of that suit.&nbsp;<em>S</em>&nbsp;= spades,&nbsp;<em>H</em>&nbsp;= Hearts,&nbsp;<em>D</em>&nbsp;= Diamonds,&nbsp;<em>C</em>&nbsp;= Clubs.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<ol style=\"list-style-type:lower-alpha\">\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">Suppose you pick four cards, but do not put any cards back into the deck. Your cards are&nbsp;<em>QS</em>, 1<em>D</em>, 1<em>C</em>,&nbsp;<em>QD</em>.</span></span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">Suppose you pick four cards and put each card back before you pick the next card. Your cards are&nbsp;<em>KH</em>, 7<em>D</em>, 6<em>D</em>,&nbsp;<em>KH</em>.</span></span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">Which of a. or b. did you sample with replacement and which did you sample without replacement?</span></span></p>\n	</li>\n</ol>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><strong>Solution&nbsp;3.5</strong></span></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">a. Because you do not put any cards back, the deck changes after each draw. These events are dependent, and this is sampling without replacement; b. Because you put each card back before picking the next one, the deck never changes. These events are independent, so this is sampling with replacement.</span></span></p>','','1','3.2.1',NULL,'1','2','1','0','1','2021-06-23 06:06:53','2021-06-23 11:36:53'),(2265,97,'3.2.2 Mutually Exclusive Events','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><em><span style=\"color:black\">A</span></em><span style=\"color:black\">&nbsp;and&nbsp;<em>B</em>&nbsp;are&nbsp;<strong>mutually exclusive</strong>&nbsp;events if they cannot occur at the same time. This means that&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;do not share any outcomes and&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) = 0.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">For example, suppose the sample space&nbsp;<em>S</em>&nbsp;= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Let&nbsp;<em>A</em>&nbsp;= {1, 2, 3, 4, 5},&nbsp;<em>B</em>&nbsp;= {4, 5, 6, 7, 8}, and&nbsp;<em>C</em>&nbsp;= {7, 9}.&nbsp;<em>A</em>&nbsp;AND&nbsp;<em>B</em>&nbsp;= {4, 5}.&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) =&nbsp;210210&nbsp;and is not equal to zero. Therefore,&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;are not mutually exclusive.</span></span></span></p>\n\n<p style=\"text-align:justify\"><br />\n<span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\"><em>A</em>&nbsp;and&nbsp;<em>C</em>&nbsp;do not have any numbers in common so&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>C</em>) = 0. Therefore,&nbsp;<em>A</em>&nbsp;and&nbsp;<em>C</em>&nbsp;are mutually exclusive.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">If it is not known whether&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;are mutually exclusive,&nbsp;<strong>assume they are not until you can show otherwise</strong>. The following examples illustrate these definitions and terms.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>EXAMPLE&nbsp;3.6</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Flip two fair coins. This is an experiment.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The sample space is {<em>HH</em>,&nbsp;<em>HT</em>,&nbsp;<em>TH</em>,&nbsp;<em>TT</em>}, where&nbsp;<em>T</em>&nbsp;= tails and&nbsp;<em>H</em>&nbsp;= heads. The outcomes are&nbsp;<em>HH</em>,&nbsp;<em>HT</em>,&nbsp;<em>TH</em>, and&nbsp;<em>TT</em>. The outcomes&nbsp;<em>HT</em>&nbsp;and&nbsp;<em>TH</em>&nbsp;are different. The&nbsp;<em>HT</em>&nbsp;means that the first coin showed heads and the second coin showed tails. The&nbsp;<em>TH</em>&nbsp;means that the first coin showed tails and the second coin showed heads.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<ul>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">Let&nbsp;<em>A</em>&nbsp;= the event of getting&nbsp;<strong>at most one tail</strong>. At most one tail means zero or one tail. Then&nbsp;<em>A</em>&nbsp;can be written as {<em>HH</em>,&nbsp;<em>HT</em>,&nbsp;<em>TH</em>}. The outcome&nbsp;<em>HH</em>&nbsp;shows zero tails.&nbsp;<em>HT</em>&nbsp;and&nbsp;<em>TH</em>&nbsp;each show one tail.</span></span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">Let&nbsp;<em>B</em>&nbsp;= the event of getting all tails.&nbsp;<em>B</em>&nbsp;can be written as {<em>TT</em>}.&nbsp;<em>B</em>&nbsp;is the&nbsp;<strong>complement event</strong>&nbsp;of&nbsp;<em>A</em>, so&nbsp;<em>B</em>&nbsp;=&nbsp;<em>A&prime;</em>. Also,&nbsp;<em>P</em>(<em>A</em>) +&nbsp;<em>P</em>(<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>) +&nbsp;<em>P</em>(<em>A&prime;</em>) = 1.</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The probabilities for&nbsp;<em>A</em>&nbsp;and for&nbsp;<em>B</em>&nbsp;are&nbsp;</span><span class=\"math-tex\">\\(P(A)=\\frac{3}{4}\\hspace{0.1cm}and\\hspace{0.1cm}P(B)=\\frac{1}{4}\\)</span></span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">Let&nbsp;<em>C</em>&nbsp;= the event of getting all heads.&nbsp;<em>C</em>&nbsp;= {<em>HH</em>}. Since&nbsp;<em>B</em>&nbsp;= {<em>TT</em>},&nbsp;<em>P</em>(<em>B</em>&nbsp;AND&nbsp;<em>C</em>) = 0.&nbsp;<em>B</em>&nbsp;and&nbsp;<em>C</em>&nbsp;are mutually exclusive. (<em>B</em>&nbsp;and&nbsp;<em>C</em>&nbsp;have no members in common because you cannot have all tails and all heads at the same time.)</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Let&nbsp;<em>D</em>&nbsp;= event of getting&nbsp;<strong>more than one</strong>&nbsp;tail.&nbsp;<em>D</em>&nbsp;= {<em>TT</em>}.&nbsp;<span class=\"math-tex\">\\(P(D)=\\frac{1}{4}\\)</span></span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Let&nbsp;<em>E</em>&nbsp;= event of getting a head on the first roll. This implies you can get either a head or tail on the second roll.&nbsp;<em>E</em>&nbsp;= {<em>HT</em>,&nbsp;<em>HH</em>}.&nbsp;<span class=\"math-tex\">\\(P(E)=\\frac{2}{4}\\)</span></span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Find the probability of getting&nbsp;<strong>at least one</strong>&nbsp;(one or two) tail in two flips. Let&nbsp;<em>F</em>&nbsp;= event of getting at least one tail in two flips.&nbsp;<em>F</em>&nbsp;= {<em>HT</em>,&nbsp;<em>TH</em>,&nbsp;<em>TT</em>}.&nbsp;<span class=\"math-tex\">\\(P(F)=\\frac{3}{4}\\)</span></span></span></p>\n	</li>\n</ul>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>EXAMPLE&nbsp;3.7</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Flip two fair coins. Find the probabilities of the events.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<ol style=\"list-style-type:lower-alpha\">\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\">Let&nbsp;<em>F</em>&nbsp;= the event of getting at most one tail (zero or one tail).</span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\">Let&nbsp;<em>G</em>&nbsp;= the event of getting two faces that are the same.</span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\">Let&nbsp;<em>H</em>&nbsp;= the event of getting a head on the first flip followed by a head or tail on the second flip.</span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\">Are&nbsp;<em>F</em>&nbsp;and&nbsp;<em>G</em>&nbsp;mutually exclusive?</span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\">Let&nbsp;<em>J</em>&nbsp;= the event of getting all tails. Are&nbsp;<em>J</em>&nbsp;and&nbsp;<em>H</em>&nbsp;mutually exclusive?</span></p>\n	</li>\n</ol>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong><span style=\"color:#002060\">Solution&nbsp;3.7</span></strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Look at the sample space in&nbsp;</span><a href=\"https://openstax.org/books/statistics/pages/3-2-independent-and-mutually-exclusive-events#element-931\" style=\"color:blue; text-decoration:underline\"><span style=\"color:#027eb5\">Example 3.6</span></a><span style=\"color:#424242\">.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<ol start=\"1\" style=\"list-style-type:lower-alpha\">\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Zero (0) or one (1) tails occur when the outcomes&nbsp;<em>HH</em>,&nbsp;<em>TH</em>,&nbsp;<em>HT</em>&nbsp;show up.&nbsp;<span class=\"math-tex\">\\(P(F)=\\frac{3}{4}\\)</span></span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Two faces are the same if&nbsp;<em>HH</em>&nbsp;or&nbsp;<em>TT</em>&nbsp;show up.&nbsp;<span class=\"math-tex\">\\(P(G)=\\frac{2}{4}\\)</span></span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">A head on the first flip followed by a head or tail on the second flip occurs when&nbsp;<em>HH</em>&nbsp;or&nbsp;<em>HT</em>&nbsp;show up.&nbsp;<span class=\"math-tex\">\\(P(H)=\\frac{2}{4}\\)</span></span></span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\"><em>F</em>&nbsp;and&nbsp;<em>G</em>&nbsp;share&nbsp;<em>HH</em>&nbsp;so&nbsp;<em>P</em>(<em>F</em>&nbsp;AND&nbsp;<em>G</em>) is not equal to zero (0).&nbsp;<em>F</em>&nbsp;and&nbsp;<em>G</em>&nbsp;are not mutually exclusive.</span></span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">Getting all tails occurs when tails shows up on both coins (<em>TT</em>).&nbsp;<em>H</em>&rsquo;s outcomes are&nbsp;<em>HH</em>&nbsp;and&nbsp;<em>HT</em>.</span></span></p>\n	</li>\n</ol>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">J</span></em><span style=\"color:black\">&nbsp;and&nbsp;<em>H</em>&nbsp;have nothing in common so&nbsp;<em>P</em>(<em>J</em>&nbsp;AND&nbsp;<em>H</em>) = 0.&nbsp;<em>J</em>&nbsp;and&nbsp;<em>H</em>&nbsp;are mutually exclusive.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>EXAMPLE&nbsp;3.8</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Roll one fair, six-sided die. The sample space is {1, 2, 3, 4, 5, 6}. Let event&nbsp;<em>A</em>&nbsp;= a face is odd. Then&nbsp;<em>A</em>&nbsp;= {1, 3, 5}. Let event&nbsp;<em>B</em>&nbsp;= a face is even. Then&nbsp;<em>B</em>&nbsp;= {2, 4, 6}.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Find the complement of&nbsp;<em>A</em>,&nbsp;<em>A&prime;</em>. The complement of&nbsp;<em>A</em>,&nbsp;<em>A&prime;</em>, is&nbsp;<em>B</em>&nbsp;because&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;together make up the sample space.&nbsp;<em>P</em>(<em>A</em>) +&nbsp;<em>P</em>(<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>) +&nbsp;<em>P</em>(<em>A&prime;</em>) = 1. Also,&nbsp;</span><span class=\"math-tex\">\\(P(A)=\\frac{3}{6}\\hspace{0.1cm}and\\hspace{0.1cm}P(B)=\\frac{3}{6}\\)</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<ul>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">Let event&nbsp;<em>C</em>&nbsp;= odd faces larger than two. Then&nbsp;<em>C</em>&nbsp;= {3, 5}. Let event&nbsp;<em>D</em>&nbsp;= all even faces smaller than five. Then&nbsp;<em>D</em>&nbsp;= {2, 4}.&nbsp;<em>P</em>(<em>C</em>&nbsp;AND&nbsp;<em>D</em>) = 0 because you cannot have an odd and even face at the same time. Therefore,&nbsp;<em>C</em>&nbsp;and&nbsp;<em>D</em>&nbsp;are mutually exclusive events.</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Let event&nbsp;<em>E</em>&nbsp;= all faces less than five.&nbsp;<em>E</em>&nbsp;= {1, 2, 3, 4}.</span></span></p>\n	</li>\n</ul>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Are&nbsp;<em>C</em>&nbsp;and&nbsp;<em>E</em>&nbsp;mutually exclusive events? Answer yes or no. Why or why not?</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>Solution&nbsp;3.8</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<ul>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">No.&nbsp;<em>C</em>&nbsp;= {3, 5} and&nbsp;<em>E</em>&nbsp;= {1, 2, 3, 4}.&nbsp;<em>P</em>(<em>C</em>&nbsp;AND&nbsp;<em>E</em>) </span><span class=\"math-tex\">\\(=\\frac{1}{6}\\)</span></span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">&nbsp;&nbsp;To be mutually exclusive,&nbsp;<em>P</em>(<em>C</em>&nbsp;AND&nbsp;<em>E</em>) must be zero.</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Find&nbsp;<em>P</em>(<em>C</em>|<em>A</em>). This is a conditional probability. Recall that event&nbsp;<em>C</em>&nbsp;is {3, 5} and event&nbsp;<em>A</em>&nbsp;is {1, 3, 5}. To find&nbsp;<em>P</em>(<em>C</em>|<em>A</em>), find the probability of&nbsp;<em>C</em>&nbsp;using the sample space&nbsp;<em>A</em>. You have reduced the sample space from the original sample space {1, 2, 3, 4, 5, 6} to {1, 3, 5}. So,&nbsp;<em>P</em>(<em>C</em>|<em>A</em>)=<span class=\"math-tex\">\\(\\frac{2}{3}\\)</span></span></span></p>\n	</li>\n</ul>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong><span style=\"color:#002060\">EXAMPLE&nbsp;3.9</span></strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Let event&nbsp;<em>G</em>&nbsp;= taking a math class. Let event&nbsp;<em>H</em>&nbsp;= taking a science class. Then,&nbsp;<em>G</em>&nbsp;AND&nbsp;<em>H</em>&nbsp;= taking a math class and a science class. Suppose&nbsp;<em>P</em>(<em>G</em>) = .6,&nbsp;<em>P</em>(<em>H</em>) = .5, and&nbsp;<em>P</em>(<em>G</em>&nbsp;AND&nbsp;<em>H</em>) = .3. Are&nbsp;<em>G</em>&nbsp;and&nbsp;<em>H</em>&nbsp;independent?</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">If&nbsp;<em>G</em>&nbsp;and&nbsp;<em>H</em>&nbsp;are independent, then you must show&nbsp;<strong>ONE</strong>&nbsp;of the following:</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<ul>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\"><em>P</em>(<em>G</em>|<em>H</em>) =&nbsp;<em>P</em>(<em>G</em>)</span></span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\"><em>P</em>(<em>H</em>|<em>G</em>) =&nbsp;<em>P</em>(<em>H</em>)</span></span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\"><em>P</em>(<em>G</em>&nbsp;AND&nbsp;<em>H</em>) =&nbsp;<em>P</em>(<em>G</em>)<em>P</em>(<em>H</em>)</span></span></p>\n	</li>\n</ul>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">a. Show that&nbsp;<em>P</em>(<em>G</em>|<em>H</em>) =&nbsp;<em>P</em>(<em>G</em>).</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><strong><span style=\"font-size:14px\"><span style=\"color:#002060\">Solution&nbsp;3.9</span></span></strong></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\"><span class=\"math-tex\">\\(P(G\\H)=\\frac{P(G\\hspace{0.1cm}AND\\hspace{0.1cm}H)}{P(H)}=\\frac{3}{5}=6=P(G)\\)</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">. Show&nbsp;<em>P</em>(<em>G</em>&nbsp;AND&nbsp;<em>H</em>) =&nbsp;<em>P</em>(<em>G</em>)<em>P</em>(<em>H</em>).</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><strong><span style=\"font-size:14px\"><span style=\"color:#002060\">Solution&nbsp;3.9</span></span></strong></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:#424242\">P</span></em>(<em><span style=\"color:#424242\">G</span></em>)<em><span style=\"color:#424242\">P</span></em>(<em><span style=\"color:#424242\">H</span></em>) = (.6)(.5) = .3 =&nbsp;<em><span style=\"color:#424242\">P</span></em>(<em><span style=\"color:#424242\">G</span></em>&nbsp;AND&nbsp;<em><span style=\"color:#424242\">H</span></em>)</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Since&nbsp;<em>G</em>&nbsp;and&nbsp;<em>H</em>&nbsp;are independent, knowing that a person is taking a science class does not change the chance that he or she is taking a math class. If the two events had not been independent, that is, they are dependent, then knowing that a person is taking a science class would change the chance he or she is taking math. For practice, show that&nbsp;<em>P</em>(<em>H</em>|<em>G</em>) =&nbsp;<em>P</em>(<em>H</em>) to show that&nbsp;<em>G</em>&nbsp;and&nbsp;<em>H</em>&nbsp;are independent events.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>EXAMPLE&nbsp;3.10</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Let event&nbsp;<em>C</em>&nbsp;= taking an English class. Let event&nbsp;<em>D</em>&nbsp;= taking a speech class.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Suppose&nbsp;<em>P</em>(<em>C</em>) = .75,&nbsp;<em>P</em>(<em>D</em>) = .3,&nbsp;<em>P</em>(<em>C</em>|<em>D</em>) = .75 and&nbsp;<em>P</em>(<em>C</em>&nbsp;AND&nbsp;<em>D</em>) = .225.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Justify your answers to the following questions numerically.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<ol style=\"list-style-type:lower-alpha\">\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Are&nbsp;<em>C</em>&nbsp;and&nbsp;<em>D</em>&nbsp;independent?</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Are&nbsp;<em>C</em>&nbsp;and&nbsp;<em>D</em>&nbsp;mutually exclusive?</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">What is&nbsp;<em>P</em>(<em>D</em>|<em>C</em>)?</span></span></p>\n	</li>\n</ol>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution&nbsp;3.10</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<ol style=\"list-style-type:lower-alpha\">\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Yes, because&nbsp;<em>P</em>(<em>C</em>|<em>D</em>) = .75 =&nbsp;<em>P</em>(<em>C</em>).</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">No, because&nbsp;<em>P</em>(<em>C</em>&nbsp;AND&nbsp;<em>D</em>) is not equal to zero.</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">D</span></em><span style=\"color:black\">|</span><em><span style=\"color:black\">C</span></em><span style=\"color:black\">) </span><span class=\"math-tex\">\\(=\\frac{P(C\\hspace{0.1cm}AND\\hspace{0.1cm}D)}{P(C)}=\\frac{0.225}{0.750}=.3\\)</span></span></p>\n	</li>\n</ol>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>EXAMPLE&nbsp;3.11</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<ol style=\"list-style-type:lower-alpha\">\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">In a box there are three red cards and five blue cards. The red cards are marked with the numbers 1, 2, and 3, and the blue cards are marked with the numbers 1, 2, 3, 4, and 5. The cards are well-shuffled. You reach into the box (you cannot see into it) and draw one card.</span></span></p>\n	</li>\n</ol>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"margin-left:48px; text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Let&nbsp;<em>R</em>&nbsp;= red card is drawn,&nbsp;<em>B</em>&nbsp;= blue card is drawn,&nbsp;<em>E</em>&nbsp;= even-numbered card is drawn.</span></span></p>\n\n<p style=\"margin-left:48px; text-align:justify\">&nbsp;</p>\n\n<p style=\"margin-left:48px; text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The sample space&nbsp;<em>S</em>&nbsp;=&nbsp;<em>R</em>1,&nbsp;<em>R</em>2,&nbsp;<em>R</em>3,&nbsp;<em>B</em>1,&nbsp;<em>B</em>2,&nbsp;<em>B</em>3,&nbsp;<em>B</em>4,&nbsp;<em>B</em>5.&nbsp;<em>S</em>&nbsp;has eight outcomes.</span></span></p>\n\n<p style=\"margin-left:48px; text-align:justify\">&nbsp;</p>\n\n<ul>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\"><span class=\"math-tex\">\\(P(R)=\\frac{3}{8}. P(B)=\\frac{5}{8}\\)</span></span>&nbsp;&nbsp;<em><span style=\"color:black\">P</span></em><span style=\"color:black\">(<em>R</em>&nbsp;AND&nbsp;<em>B</em>) = 0. You cannot draw one card that is both red and blue.</span></span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\"><span class=\"math-tex\">\\(P(E)=\\frac{3}{8}. \\)</span>There are three even-numbered cards,&nbsp;<em>R</em>2,&nbsp;<em>B</em>2, and&nbsp;<em>B</em>4.</span></span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(<em>E</em>|<em>B</em>&nbsp;</span>)<span class=\"math-tex\">\\(=\\frac{2}{5}\\)</span>&nbsp;<span style=\"color:black\">. There are five blue cards:&nbsp;<em>B</em>1,&nbsp;<em>B</em>2,&nbsp;<em>B</em>3,&nbsp;<em>B</em>4, and&nbsp;<em>B</em>5. Out of the blue cards, there are two even cards;&nbsp;<em>B</em>2 and&nbsp;<em>B</em>4.</span></span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(<em>B</em>|<em>E</em>)&nbsp;<span class=\"math-tex\">\\(=\\frac{2}{3}\\)</span></span>&nbsp;</span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">. There are three even-numbered cards:&nbsp;<em>R</em>2,&nbsp;<em>B</em>2, and&nbsp;<em>B</em>4. Out of the even-numbered cards, two are blue;&nbsp;<em>B</em>2 and&nbsp;<em>B</em>4.</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The events&nbsp;<em>R</em>&nbsp;and&nbsp;<em>B</em>&nbsp;are mutually exclusive because&nbsp;<em>P</em>(<em>R</em>&nbsp;AND&nbsp;<em>B</em>) = 0.</span></span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">Let&nbsp;<em>G</em>&nbsp;= card with a number greater than 3.&nbsp;<em>G</em>&nbsp;= {<em>B</em>4,&nbsp;<em>B</em>5}.&nbsp;<em>P</em>(<em>G</em>)&nbsp;<span class=\"math-tex\">\\(=\\frac{2}{8}\\)</span>&nbsp;Let&nbsp;<em>H</em>&nbsp;= blue card numbered between one and four, inclusive.&nbsp;<em>H</em>&nbsp;= {<em>B</em>1,&nbsp;<em>B</em>2,&nbsp;<em>B</em>3,&nbsp;<em>B</em>4}.&nbsp;<em>P</em>(<em>G</em>|<em>H</em>)=&nbsp;<span class=\"math-tex\">\\(\\frac{1}{4}\\)</span>&nbsp;. The only card in&nbsp;<em>H</em>&nbsp;that has a number greater than three is&nbsp;<em>B</em>4. Since&nbsp;<span class=\"math-tex\">\\(=\\frac{2}{8}=\\frac{1}{4}\\)</span>&nbsp;</span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(<em>G</em>) =&nbsp;<em>P</em>(<em>G</em>|<em>H</em>), which means that&nbsp;<em>G</em>&nbsp;and&nbsp;<em>H</em>&nbsp;are independent.</span></span></p>\n	</li>\n</ul>\n\n<p style=\"margin-left:48px; text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">??????????????</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>EXAMPLE&nbsp;3.12</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">In a particular class, 60 percent of the students are female. Fifty percent of all students in the class have long hair. Forty-five percent of the students are female and have long hair. Of the female students, 75 percent have long hair. Let&nbsp;<em>F</em>&nbsp;be the event that a student is female. Let&nbsp;<em>L</em>&nbsp;be the event that a student has long hair. One student is picked randomly. Are the events of being female and having long hair independent?</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">The following probabilities are given in this example:</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<ul>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\"><em>P</em>(<em>F</em>) = 0.60;&nbsp;<em>P</em>(<em>L</em>) = 0.50</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\"><em>P</em>(<em>F</em>&nbsp;AND&nbsp;<em>L</em>) = 0.45</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\"><em>P</em>(<em>L</em>|<em>F</em>) = 0.75</span></span></p>\n	</li>\n</ul>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">??????????????</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>Solution 3.12</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:#ededed\"><span style=\"color:black\">Check whether&nbsp;</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">P</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">(</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">F</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">&nbsp;AND&nbsp;</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">L</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">) =&nbsp;</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">P</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">(</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">F</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">)</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">P</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">(</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">L</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">). We are given that&nbsp;</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">P</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">(</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">F</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">&nbsp;AND&nbsp;</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">L</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">) = 0.45, but&nbsp;</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">P</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">(</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">F</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">)</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">P</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">(</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">L</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">) = (.60)(.50) = .30. The events of being female and having long hair are not independent because&nbsp;</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">P</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">(</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">F</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">&nbsp;AND&nbsp;</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">L</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">) does not equal&nbsp;</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">P</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">(</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">F</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">)</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">P</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">(</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">L</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">).</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>Solution 3.12</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:#ededed\"><span style=\"color:black\">Check whether&nbsp;</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">P</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">(</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">L</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">|</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">F</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">) equals&nbsp;</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">P</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">(</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">L</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">). We are given that&nbsp;</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">P</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">(</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">L</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">|</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">F</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">) = .75, but&nbsp;</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">P</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">(</span></span><em><span style=\"background-color:#ededed\"><span style=\"color:#424242\">L</span></span></em><span style=\"background-color:#ededed\"><span style=\"color:black\">) = .50; they are not equal. The events of being female and having long hair are not independent.</span></span></span></p>\n\n<p style=\"text-align:justify\"><br />\n<br />\n<strong><span style=\"font-size:14px\">Interpretation of Results</span></strong></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">The events of being female and having long hair are not independent; knowing that a student is female changes the probability that a student has long hair.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\">EXAMPLE&nbsp;3.13</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<ol style=\"list-style-type:lower-alpha\">\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Toss one fair coin (the coin has two sides,&nbsp;<em>H</em>&nbsp;and&nbsp;<em>T</em>). The outcomes are ________. Count the outcomes. There are ________ outcomes.</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Toss one fair, six-sided die (the die has 1, 2, 3, 4, 5, or 6 dots on a side). The outcomes are ________. Count the outcomes. There are ________ outcomes.</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Multiply the two numbers of outcomes. The answer is ________.</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">If you flip one fair coin and follow it with the toss of one fair, six-sided die, the answer in Part c is the number of outcomes (size of the sample space). List the outcomes. Hint&mdash;Two of the outcomes are&nbsp;<em>H</em>1 and&nbsp;<em>T</em>6.</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Event&nbsp;<em>A</em>&nbsp;= heads (<em>H</em>) on the coin followed by an even number (2, 4, 6) on the die.<br />\n	<em>A</em>&nbsp;= {________}. Find&nbsp;<em>P</em>(<em>A</em>).</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Event&nbsp;<em>B</em>&nbsp;= heads on the coin followed by a three on the die.&nbsp;<em>B</em>&nbsp;= {________}. Find&nbsp;<em>P</em>(<em>B</em>).</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Are&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;mutually exclusive? Hint&mdash;What is&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>)? If&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) = 0, then&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;are mutually exclusive.</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Are&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;independent? Hint&mdash;Is&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>)<em>P</em>(<em>B</em>)? If&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>)<em>P</em>(<em>B</em>), then&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;are independent. If not, then they are dependent.</span></span></p>\n	</li>\n</ol>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">???????</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution&nbsp;3.13</strong></span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>???????</strong></span></span></p>\n\n<ol style=\"list-style-type:lower-alpha\">\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">H</span></em><span style=\"color:black\">&nbsp;and&nbsp;<em>T</em>; 2</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">1, 2, 3, 4, 5, 6; 6</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">2(6) = 12</span></span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">Make a systematic list of possible outcomes. Start by listing all possible outcomes when the coin shows tails (<em>T</em>). Then list the outcomes that are possible when the coin shows heads (<em>H</em>):&nbsp;<em>T</em>1,&nbsp;<em>T</em>2,&nbsp;<em>T</em>3,&nbsp;<em>T</em>4,&nbsp;<em>T</em>5,&nbsp;<em>T</em>6,&nbsp;<em>H</em>1,&nbsp;<em>H</em>2,&nbsp;<em>H</em>3,&nbsp;<em>H</em>4,&nbsp;<em>H</em>5,&nbsp;<em>H</em>6</span></span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><em><span style=\"color:black\">A</span></em><span style=\"color:black\">&nbsp;= {<em>H</em>2,&nbsp;<em>H</em>4,&nbsp;<em>H</em>6};&nbsp;</span><span class=\"math-tex\">\\(P(A)=\\frac{NUMBER\\hspace{0.1cm}OF\\hspace{0.1cm}OUTCOMES\\hspace{0.1cm}IN\\hspace{0.1cm}A}{NUMBER\\hspace{0.1cm}OF\\hspace{0.1cm}POSSIBLE\\hspace{0.1cm}OUTCOMES}=\\frac{3}{12}\\)</span></span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><em><span style=\"color:black\">B</span></em><span style=\"color:black\">&nbsp;= {<em>H</em>3};&nbsp;<em>P</em>(<em>B</em>) </span><span class=\"math-tex\">\\(=\\frac{1}{2}\\)</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Yes, because&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) = 0</span></span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) = 0.&nbsp;<em>P</em>(<em>A</em>)<em>P</em>(<em>B</em>)=<span class=\"math-tex\">\\((\\frac{3}{12})(\\frac{1}{2})\\)</span>&nbsp;</span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) does not equal&nbsp;<em>P</em>(<em>A</em>)<em>P</em>(<em>B</em>), so&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;are dependent.</span></span></p>\n	</li>\n</ol>\n\n<p>&nbsp;</p>','','1','3.2.2',NULL,'1','2','1','0','1','2021-06-23 06:10:49','2021-06-23 11:40:49'),(2266,97,'3.3 Two Basic Rules of Probability','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">In calculating probability, there are two rules to consider when you are determining if two events are independent or dependent and if they are mutually exclusive or not.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>The Multiplication Rule</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">If&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;are two events defined on a&nbsp;<strong>sample space</strong>, then&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>B</em>)<em>P</em>(<em>A</em>|<em>B</em>).</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">This equation can be rewritten as&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>B</em>)<em>P</em>(<em>A</em>|<em>B</em>), the multiplication rule.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">If&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;are&nbsp;<strong>independent</strong>, then&nbsp;<em>P</em>(<em>A</em>|<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>). In this special case,&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>|<em>B</em>)<em>P</em>(<em>B</em>) becomes&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>)<em>P</em>(<em>B</em>).</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">A bag contains four green marbles, three red marbles, and two yellow marbles. Mark draws two marbles from the bag without replacement. The probability that he draws a yellow marble and then a green marble is&nbsp;</span></span><span style=\"font-size:14px\"><span style=\"color:black\">P(yellow&nbsp;and&nbsp;green)=P(yellow)&sdot;P(green&nbsp;|&nbsp;yellow)<span class=\"math-tex\">\\(\\frac{2}{9},\\frac{4}{8}\\)</span></span></span><span style=\"font-size:14px\"><span style=\"color:black\">. Notice that&nbsp;P(green | yellow)=<span class=\"math-tex\">\\(\\frac{4}{8}\\)</span></span>&nbsp;.&nbsp;</span><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">After the yellow marble is drawn, there are four green marbles in the bag and eight marbles in all.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>The Addition Rule</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">If&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;are defined on a sample space, then&nbsp;<em>P</em>(<em>A</em>&nbsp;OR&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>) +&nbsp;<em>P</em>(<em>B</em>) &minus;&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>).</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">Draw one card from a standard deck of playing cards. Let&nbsp;<em>H</em>&nbsp;= the card is a heart, and let&nbsp;<em>J</em>&nbsp;= the card is a jack. These events are not mutually exclusive because a card can be both a heart and a jack.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">P(H&nbsp;or&nbsp;J)=P(H)+P(J)&minus;P(H&nbsp;and&nbsp;J)</span></span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\"><span class=\"math-tex\">\\(=\\frac{13}{52}+\\frac{4}{52}-\\frac{1}{52}\\\\ =\\frac{16}{52} \\)</span></span></span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">&asymp;.3077</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">If&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;are&nbsp;<strong>mutually exclusive</strong>, then&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) = 0. Then&nbsp;<em>P</em>(<em>A</em>&nbsp;OR&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>) +&nbsp;<em>P</em>(<em>B</em>) &minus;&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) becomes <em>P</em>(<em>A</em>&nbsp;OR&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>) +&nbsp;<em>P</em>(<em>B</em>).</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">Draw one card from a standard deck of playing cards. Let&nbsp;<em>H</em>&nbsp;= the card is a heart and&nbsp;<em>S</em>&nbsp;= the card is a spade. These events are mutually exclusive because a card cannot be a heart and a spade at the same time. The probability that the card is a heart or a spade is</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">P(H&nbsp;or&nbsp;S)=P(H)+P(S)=</span></span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\"><span class=\"math-tex\">\\(=\\frac{13}{52}+\\frac{13}{52}\\\\ =\\frac{26}{52}\\\\ =\\frac{1}{2}\\)</span></span></span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">&asymp;.3077</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">If&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;are&nbsp;<strong>mutually exclusive</strong>, then&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) = 0. Then&nbsp;<em>P</em>(<em>A</em>&nbsp;OR&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>) +&nbsp;<em>P</em>(<em>B</em>) &minus;&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) becomes <em>P</em>(<em>A</em>&nbsp;OR&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>) +&nbsp;<em>P</em>(<em>B</em>).</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">Draw one card from a standard deck of playing cards. Let&nbsp;<em>H</em>&nbsp;= the card is a heart and&nbsp;<em>S</em>&nbsp;= the card is a spade. These events are mutually exclusive because a card cannot be a heart and a spade at the same time. The probability that the card is a heart or a spade i</span></span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">P(H&nbsp;or&nbsp;S)=P(H)+P(S)</span></span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\"><span class=\"math-tex\">\\(=\\frac{13}{52}+\\frac{13}{52}\\)</span></span></span></span></p>\n\n<p><span style=\"font-size:14px\">&nbsp;<span class=\"math-tex\">\\(=\\frac{26}{52}=\\frac{1}{2}\\)</span>&nbsp; &nbsp;=.5</span></p>','','1','3.3',NULL,'1','2','1','0','1','2021-06-23 06:12:42','2021-06-23 11:42:42'),(2267,97,'3.3.1 EXAMPLES','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>EXAMPLE&nbsp;3.14&nbsp;&nbsp; </strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Krishna is trying to choose where to go on vacation. His two choices are:&nbsp;<em>A</em>&nbsp;= New Zealand and&nbsp;<em>B</em>&nbsp;= Alaska.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<ul>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">Krishna can only afford one vacation. The probability that he chooses&nbsp;<em>A</em>&nbsp;is&nbsp;<em>P</em>(<em>A</em>) = .6 and the probability that he chooses&nbsp;<em>B</em>&nbsp;is&nbsp;<em>P</em>(<em>B</em>) = .35.</span></span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\"><em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) = 0 because Klaus can only afford to take one vacation.</span></span></p>\n	</li>\n	<li style=\"text-align:justify\">\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">Therefore, the probability that he chooses either New Zealand or Alaska is&nbsp;<em>P</em>(<em>A</em>&nbsp;OR&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>) +&nbsp;<em>P</em>(<em>B</em>) = .6 + .35 = .95. Note that the probability that he does not choose to go anywhere on vacation must be .05.</span></span></p>\n	</li>\n</ul>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>EXAMPLE&nbsp;3.15</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Carlos plays college soccer. He makes a goal 65 percent of the time he shoots. Carlos is going to attempt two goals in a row in the next game.&nbsp;<em>A</em>&nbsp;= the event Carlos is successful on his first attempt.&nbsp;<em>P</em>(<em>A</em>) = .65.&nbsp;<em>B</em>&nbsp;= the event Carlos is successful on his second attempt.&nbsp;<em>P</em>(<em>B</em>) = .65. Carlos tends to shoot in streaks. The probability that he makes the second goal&nbsp;<strong>given</strong>&nbsp;that he made the first goal is .90.</span><br />\n<span style=\"color:#424242\">a. </span><span style=\"color:black\">What is the probability that he makes both goals?</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution&nbsp;3.15</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>Solution&nbsp;</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">a. The problem is asking you to find&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>B</em>&nbsp;AND&nbsp;<em>A</em>). Since&nbsp;<em>P</em>(<em>B</em>|<em>A</em>) = :&nbsp;<em>P</em>(<em>B</em>&nbsp;AND&nbsp;<em>A</em>) =&nbsp;<em>P</em>(<em>B</em>|<em>A</em>)&nbsp;<em>P</em>(<em>A</em>)&nbsp;=&nbsp;(.90)(.65) = .585.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Carlos makes the first and second goals with probability .585.<br />\nb. What is the probability that Carlos makes either the first goal or the second goal?</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>Solution&nbsp;</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">b.The problem is asking you to find&nbsp;<em>P</em>(<em>A</em>&nbsp;OR&nbsp;<em>B</em>).<em> </em></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"margin-left:24px; text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(<em>A</em>&nbsp;OR&nbsp;<em>B</em>) =&nbsp;<em>P</em>(<em>A</em>) +&nbsp;<em>P</em>(<em>B</em>) &minus;&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) = .65 + .65 &minus; .585 = .715</span></span></p>\n\n<p style=\"margin-left:24px; text-align:justify\">&nbsp;</p>\n\n<div>\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Carlos makes either the first goal or the second goal with probability .715.</span></span></p>\n</div>\n\n<p style=\"text-align:justify\"><br />\n<br />\n<span style=\"font-size:14px\"><span style=\"color:black\">c. Are&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;independent? Solution&nbsp;3</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>Solution&nbsp;</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">c.No, they are not, because&nbsp;<em>P</em>(<em>B</em>&nbsp;AND&nbsp;<em>A</em>) = .585.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(<em>B</em>)<em>P</em>(<em>A</em>) = (.65)(.65) = .423</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">.423 &ne; .585 =&nbsp;<em>P</em>(<em>B</em>&nbsp;AND&nbsp;<em>A</em>)</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">So,&nbsp;<em>P</em>(<em>B</em>&nbsp;AND&nbsp;<em>A</em>) is&nbsp;<strong>not</strong>&nbsp;equal to&nbsp;<em>P</em>(<em>B</em>)<em>P</em>(<em>A</em>).</span></span><br />\n<br />\n&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#424242\">d. </span>Are&nbsp;<em>A</em>&nbsp;and&nbsp;<em>B</em>&nbsp;mutually exclusive?</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>Solution&nbsp;</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">d. No, they are not because&nbsp;<em>P</em>(<em>A</em>&nbsp;and&nbsp;<em>B</em>) = .585.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">To be mutually exclusive,&nbsp;<em>P</em>(<em>A</em>&nbsp;AND&nbsp;<em>B</em>) must equal zero.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>EXAMPLE&nbsp;3.16</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">A community swim team has&nbsp;150&nbsp;members.&nbsp;Seventy-five&nbsp;of the members are advanced swimmers.&nbsp;Forty-seven&nbsp;of the members are intermediate swimmers. The remainder are novice swimmers.&nbsp;Forty&nbsp;of the advanced swimmers practice four times a week.&nbsp;Thirty&nbsp;of the intermediate swimmers practice four times a week.&nbsp;Ten&nbsp;of the novice swimmers practice four times a week. Suppose<span style=\"color:#424242\"> one member of the swim team is chosen randomly.</span></span></p>\n\n<p style=\"text-align:justify\"><br />\n<span style=\"font-size:14px\"><span style=\"color:#424242\">a. </span><span style=\"color:black\">What is the probability that the member is a novice swimmer?</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution&nbsp;a</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">a. There are&nbsp;<strong>150</strong>&nbsp;members; 75 of these are advanced, and 47 of these are intermediate swimmers. So there are 150 &minus; 75 &minus; 47 =&nbsp;<strong>28</strong>&nbsp;novice swimmers. The probability that a randomly selected swimmer is a novice is&nbsp;28150.28150.</span></span></p>\n\n<p style=\"text-align:justify\"><br />\n<br />\n<span style=\"font-size:14px\"><span style=\"color:#424242\">b. What is the probability that the member practices four times a week?</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution&nbsp;b</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#424242\">b.&nbsp;</span><span style=\"color:black\">40 + 30 + 10150=80150</span></span><br />\n<br />\n&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">c. What is the probability that the member is an advanced swimmer and practices four times a week?</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution&nbsp;c</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">c. There are 40 advanced swimmers who practice four times per week, so the probability is&nbsp;</span><span class=\"math-tex\">\\(\\frac{40}{150}\\)</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">d. What is the probability that a member is an advanced swimmer and an intermediate swimmer? Are being an advanced swimmer and being an intermediate swimmer mutually exclusive? Why or why not?</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution&nbsp;d</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(advanced AND intermediate)&nbsp;=&nbsp;0, so these are mutually exclusive events. A swimmer cannot be an advanced swimmer and an intermediate swimmer at the same time.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><br />\n<span style=\"font-size:14px\"><span style=\"color:black\">e. Are being a novice swimmer and practicing four times a week independent events? Why or why not?</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution&nbsp;e</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">e. No, these are not independent events.<br />\n<em>P</em>(novice AND practices four times per week) = .0667<br />\n<em>P</em>(novice)<em>P</em>(practices four times per week) = .0996<br />\n.0667 &ne; .0996</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>EXAMPLE&nbsp;3.17</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Felicity attends a school in Modesto, CA. The probability that Felicity enrolls in a math class is .2 and the probability that she enrolls in a speech class is .65. The probability that she enrolls in a math class GIVEN that she enrolls in speech class is .25.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Let&nbsp;<em>M</em>&nbsp;= math class,&nbsp;<em>S</em>&nbsp;= speech class, and&nbsp;<em>M</em>|<em>S</em>&nbsp;= math given speech.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<ol style=\"list-style-type:lower-alpha\">\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">What is the probability that Felicity enrolls in math and speech?<br />\n	Find&nbsp;<em>P</em>(<em>M</em>&nbsp;AND&nbsp;<em>S</em>) =&nbsp;<em>P</em>(<em>M</em>|<em>S</em>)<em>P</em>(<em>S</em>).</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">What is the probability that Felicity enrolls in math or speech classes?<br />\n	Find&nbsp;<em>P</em>(<em>M</em>&nbsp;OR&nbsp;<em>S</em>) =&nbsp;<em>P</em>(<em>M</em>) +&nbsp;<em>P</em>(<em>S</em>) &minus;&nbsp;<em>P</em>(<em>M</em>&nbsp;AND&nbsp;<em>S</em>).</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Are&nbsp;<em>M</em>&nbsp;and&nbsp;<em>S</em>&nbsp;independent? Is&nbsp;<em>P</em>(<em>M</em>|<em>S</em>) =&nbsp;<em>P</em>(<em>M</em>)?</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Are&nbsp;<em>M</em>&nbsp;and&nbsp;<em>S</em>&nbsp;mutually exclusive? Is&nbsp;<em>P</em>(<em>M</em>&nbsp;AND&nbsp;<em>S</em>) = 0?</span></span></p>\n	</li>\n</ol>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution&nbsp;3.17</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">a.&nbsp;<em>P</em>(<em>M</em>&nbsp;AND&nbsp;<em>S</em>) =&nbsp;<em>P</em>(<em>M</em>|<em>S</em>)<em>P</em>(<em>S</em>) = .25(.65) = .1625</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">b.&nbsp;<em>P</em>(<em>M</em>&nbsp;OR&nbsp;<em>S</em>) =&nbsp;<em>P</em>(<em>M</em>) +&nbsp;<em>P</em>(<em>S</em>) &minus;&nbsp;<em>P</em>(<em>M</em>&nbsp;AND&nbsp;<em>S</em>) = .2 + .65 &minus; .1625 = .6875</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">c. No,&nbsp;<em>P</em>(<em>M</em>|<em>S</em>) = .25 and&nbsp;<em>P</em>(<em>M</em>) = .2.</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">d. No,&nbsp;<em>P</em>(<em>M</em>&nbsp;AND&nbsp;<em>S</em>) = .1625.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>EXAMPLE&nbsp;3.18</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Researchers are studying one particular type of disease that affects women more often than men. Studies show that about one woman in seven (approximately 14.3 percent) who live to be 90 will develop the disease. Suppose that of those women who develop this disease, a test is negative 2 percent of the time. Also suppose that in the general population of women, the test for the disease is negative about 85 percent of the time. Let&nbsp;<em>B</em>&nbsp;= woman develops the disease and let&nbsp;<em>N</em>&nbsp;= tests negative. Suppose one woman is selected at random.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">a. What is the probability that the woman develops the disease? What is the probability that woman tests negative?</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution&nbsp;a</strong></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">a.&nbsp;<em>P</em>(<em>B</em>) = .143;&nbsp;<em>P</em>(<em>N</em>) = .85</span><br />\n<span style=\"color:black\">b. Given that the woman develops the disease, what is the probability that she tests negative?</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>Solution b</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">b. Among women who develop the disease, the test is negative 2 percent of the time, so </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">N</span></em><span style=\"color:black\">|</span><em><span style=\"color:black\">B</span></em><span style=\"color:black\">) = .02</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">c. What is the probability that the woman has the disease AND tests negative?</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>Solution c</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><em><span style=\"color:black\">c.P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">B </span></em><span style=\"color:black\">AND </span><em><span style=\"color:black\">N</span></em><span style=\"color:black\">) = </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">B</span></em><span style=\"color:black\">)</span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">N</span></em><span style=\"color:black\">|</span><em><span style=\"color:black\">B</span></em><span style=\"color:black\">) = (.143)(.02) = .0029</span></span></p>\n\n<p style=\"margin-left:24px; text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">d. What is the probability that the woman has the disease OR tests negative?</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>Solution d</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">d. </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">B </span></em><span style=\"color:black\">OR </span><em><span style=\"color:black\">N</span></em><span style=\"color:black\">) = </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">B</span></em><span style=\"color:black\">) + </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">N</span></em><span style=\"color:black\">) &minus; </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">B </span></em><span style=\"color:black\">AND </span><em><span style=\"color:black\">N</span></em><span style=\"color:black\">) = .143 + .85 &minus; .0029 = .9901</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">e. Are having the disease and testing negative independent events?</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>Solution e</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">e. No. </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">N</span></em><span style=\"color:black\">) = .85; </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">N</span></em><span style=\"color:black\">|</span><em><span style=\"color:black\">B</span></em><span style=\"color:black\">) = .02. So, </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">N</span></em><span style=\"color:black\">|</span><em><span style=\"color:black\">B</span></em><span style=\"color:black\">) does not equal </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">N</span></em><span style=\"color:black\">).</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">f. Are having the disease and testing negative mutually exclusive?</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>Solution f</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">f. No. </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">B </span></em><span style=\"color:black\">AND </span><em><span style=\"color:black\">N</span></em><span style=\"color:black\">) = .0029. For </span><em><span style=\"color:black\">B </span></em><span style=\"color:black\">and </span><em><span style=\"color:black\">N </span></em><span style=\"color:black\">to be mutually exclusive, </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">B </span></em><span style=\"color:black\">AND </span><em><span style=\"color:black\">N</span></em><span style=\"color:black\">) must be zero.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>EXAMPLE 3.19Exame</strong></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:white\">3.19</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Refer to the information in </span><strong><span style=\"color:#26397f\">Example 3.18</span></strong><span style=\"color:black\">. </span><em><span style=\"color:black\">P </span></em><span style=\"color:black\">= tests positive.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<ol style=\"list-style-type:lower-alpha\">\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Given that a woman develops the disease, what is the probability that she tests positive? Find </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">|</span><em><span style=\"color:black\">B</span></em><span style=\"color:black\">) = 1 &minus;</span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">N</span></em><span style=\"color:black\">|</span><em><span style=\"color:black\">B</span></em><span style=\"color:black\">).</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">What is the probability that a woman develops the disease and tests positive? </span></span></p>\n	</li>\n</ol>\n\n<p style=\"margin-left:24px; text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Find </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">B </span></em><span style=\"color:black\">AND </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">) =</span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">|</span><em><span style=\"color:black\">B</span></em><span style=\"color:black\">)</span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">B</span></em><span style=\"color:black\">).</span></span></p>\n\n<ol start=\"3\" style=\"list-style-type:lower-alpha\">\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">What is the probability that a woman does not develop the disease? Find </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">B&prime;</span></em><span style=\"color:black\">) = 1 &minus; </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">B</span></em><span style=\"color:black\">).</span></span></p>\n	</li>\n	<li>\n	<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">What is the probability that a woman tests positive for the disease? Find </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">) = 1 &minus; </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">N</span></em><span style=\"color:black\">).</span></span></p>\n	</li>\n</ol>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><strong>Solution 3.19</strong></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">a. </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">|</span><em><span style=\"color:black\">B</span></em><span style=\"color:black\">) = 1 &minus; </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">N</span></em><span style=\"color:black\">|</span><em><span style=\"color:black\">B</span></em><span style=\"color:black\">) = 1 &minus; .02 = .98</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">b. </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">B </span></em><span style=\"color:black\">AND </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">) = </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">|</span><em><span style=\"color:black\">B</span></em><span style=\"color:black\">)</span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">B</span></em><span style=\"color:black\">) = .98(.143) = .1401</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">c. </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">B&#39;</span></em><span style=\"color:black\">) = 1 &minus; </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">B</span></em><span style=\"color:black\">) = 1 &minus; .143 = .857</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">d. </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">) = 1 &minus; </span><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(</span><em><span style=\"color:black\">N</span></em><span style=\"color:black\">) = 1 &minus; .85 = .15</span></span></p>\n\n<p>&nbsp;</p>','','1','3.3.1',NULL,'1','3','1','0','1','2021-06-23 06:17:12','2021-06-23 11:47:12'),(2268,97,'3.4 Contingency Tables','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"background-color:white\"><span style=\"color:black\">A&nbsp;<strong>two-way table</strong>&nbsp;provides a way of portraying data that can facilitate calculating probabilities. When used to calculate probabilities, a two-way table is often called a&nbsp;<strong>contingency table</strong>. The table helps in determining conditional probabilities quite easily. The table displays sample values in relation to two different variables that may be dependent or contingent on one another. We used two-way tables in Chapters 1 and 2 to calculate marginal and conditional distributions. These tables organize data in a way that supports the calculation of relative frequency and, therefore, experimental (empirical) probability. Later on, we will use contingency tables again, but in another manner.</span></span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><strong>EXAMPLE&nbsp;3.20</strong></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">Suppose a study of speeding violations and drivers who use cell phones produced the following fictional data</span><span style=\"color:#002060\">:</span></span></p>\n\n<p>&nbsp;</p>\n\n<table cellspacing=\"0\" class=\"Table\" style=\"border-collapse:collapse; width:900px\">\n	<caption>\n	<p><span style=\"font-size:18px\"><strong>Table3.3</strong></span></p>\n	</caption>\n	<thead>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; height:57px; vertical-align:bottom\">\n			<div style=\"text-align:center\">&nbsp;</div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:57px; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Speeding Violation in the Last Year</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:57px; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>No Speeding Violation in the Last Year</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; height:57px; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Total</strong></span></div>\n			</td>\n		</tr>\n	</thead>\n	<tbody>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:41px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">Uses a cell phone while driving</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:41px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">25</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:41px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">280</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:41px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">305</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:58px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">Does not use a cell phone while driving</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:58px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">45</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:58px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">405</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:58px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">450</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none; height:25px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">Total</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:25px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">70</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:25px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">685</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; height:25px\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">755</span></div>\n			</td>\n		</tr>\n	</tbody>\n</table>\n\n<p>&nbsp;</p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#424242\">The total number of people in the sample is 755. The row totals are 305 and 450. The column totals are 70 and 685. Notice that 305 + 450 = 755 and 70 + 685 = 755.</span></span></p>\n\n<p>&nbsp;</p>\n\n<ol style=\"list-style-type:lower-alpha\">\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:#424242\">Using the table, calculate the following probabilities:<br />\n	Find&nbsp;<em>P</em>(Person uses a cell phone while driving).</span></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:#424242\">Find&nbsp;<em>P</em>(Person had no violation in the last year).</span></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:#424242\">Find&nbsp;<em>P</em>(Person had no violation in the last year&nbsp;<em>and</em>&nbsp;uses a cell phone while driving).</span></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:#424242\">Find&nbsp;<em>P</em>(Person uses a cell phone while driving&nbsp;<em>or</em>&nbsp;person had no violation in the last year).</span></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:#424242\">Find&nbsp;<em>P</em>(Person uses a cell phone while driving&nbsp;<em>given</em>&nbsp;person had a violation in the last year).</span></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:#424242\">Find&nbsp;<em>P</em>(Person had no violation last year&nbsp;<em>given</em>&nbsp;person does not use a cell phone while driving).</span></span></p>\n	</li>\n</ol>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution&nbsp;3.20</strong></span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#424242\">a. </span><span style=\"color:black\">This is the same as the marginal distribution (</span><a href=\"https://openstax.org/books/statistics/pages/1-2-data-sampling-and-variation-in-data-and-sampling\" style=\"color:blue; text-decoration:underline\"><span style=\"color:black\">Section 1.2</span></a><span style=\"color:black\">).</span></span></p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">P(Person uses a cell phone while driving)=<span class=\"math-tex\">\\(\\frac{number\\hspace{0.1cm}who\\hspace{0.1cm}had\\hspace{0.1cm}no\\hspace{0.1cm}violation}{number\\hspace{0.1cm}in\\hspace{0.1cm}study}=\\frac{685}{755}\\)</span>&nbsp;&asymp;.9073.</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">c. Find the number of participants who satisfy&nbsp;<em>both</em>&nbsp;conditions.</span></span></p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">P(Person&nbsp;had&nbsp;no&nbsp;violation&nbsp;in&nbsp;the&nbsp;last&nbsp;year&nbsp;AND&nbsp;uses&nbsp;a&nbsp;cell&nbsp;phone&nbsp;while&nbsp;driving)= number&nbsp;in&nbsp;study=<span class=\"math-tex\">\\(\\frac{number\\hspace{0.1cm}who\\hspace{0.1cm}had\\hspace{0.1cm}no\\hspace{0.1cm}violation\\hspace{0.1cm}AND\\hspace{0.1cm}uses\\hspace{0.1cm}cell\\hspace{0.1cm}phone\\hspace{0.1cm}while\\hspace{0.1cm}driving}{number\\hspace{0.1cm}in\\hspace{0.1cm}study}\\)</span></span></span></p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<span class=\"math-tex\">\\(=\\frac{280}{755}\\)</span></span></span></p>\n\n<p><span style=\"font-size:14px\">&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span style=\"color:black\">&asymp;.3709</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">d. To find this probability, you need to identify how many participants use a cell phone while driving OR have no violation in the past year OR both.</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#424242\">P(Person uses a cell phone while driving OR had no violation in the last year)</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#424242\"><span class=\"math-tex\">\\(\\frac{25+405+280}{755}=\\frac{710}{755}\\)</span></span>&nbsp;<span style=\"color:black\">&asymp;.9404</span></span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">e. This is a conditional probability. You are&nbsp;<em><span style=\"color:#424242\">given</span></em>&nbsp;that the person had no violation in the last year, so you need only consider the values in that column of data.</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:#424242\">(Person&nbsp;uses&nbsp;a&nbsp;cell&nbsp;phone&nbsp;while&nbsp;driving&nbsp;GIVEN&nbsp;the&nbsp;person&nbsp;had&nbsp;a&nbsp;violation&nbsp;in&nbsp;the&nbsp;last&nbsp;year)</span></span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\"><span class=\"math-tex\">\\(\\frac{number\\hspace{0.1cm}who\\hspace{0.1cm}used\\hspace{0.1cm}cell\\hspace{0.1cm}phone\\hspace{0.1cm}AND\\hspace{0.1cm}had\\hspace{0.1cm}a\\hspace{0.1cm}violation}{number\\hspace{0.1cm}in\\hspace{0.1cm}study\\hspace{0.1cm}who\\hspace{0.1cm}had\\hspace{0.1cm}a\\hspace{0.1cm}violation\\hspace{0.1cm}in\\hspace{0.1cm}the\\hspace{0.1cm}last\\hspace{0.1cm}year}=\\frac{25}{70}\\)</span>&nbsp;</span><span style=\"color:#424242\">&asymp;.3571</span> </span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">f. For this conditional probability, consider only values in the row labeled &ldquo;Does not use a cell phone while driving.&rdquo;</span></p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">P(Person had no violation last year GIVEN person does not use cell phone while driving)<span class=\"math-tex\">\\(=\\frac{405}{450}=.9\\)</span></span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>EXAMPLE&nbsp;3.21</strong></span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><a href=\"https://openstax.org/books/statistics/pages/3-4-contingency-tables#M05_ch03-tbl003\" style=\"color:blue; text-decoration:underline\"><span style=\"color:#027eb5\">Table 3.5</span></a><span style=\"color:#424242\">&nbsp;shows a random sample of 100 hikers and the areas of hiking they prefer.</span></span></p>\n\n<table cellspacing=\"0\" class=\"Table\" style=\"border-collapse:collapse; border:none; width:900px\">\n	<caption>\n	<p><span style=\"font-size:18px\"><strong>Table3.5 Hiking Area Preference</strong></span></p>\n	</caption>\n	<thead>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">Sex</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">The Coastline</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">Near Lakes and Streams</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">On Mountain Peaks</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">Total</span></span></div>\n			</td>\n		</tr>\n	</thead>\n	<tbody>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">Female</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">18</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">16</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">___</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">45</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">Male</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">___</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">___</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">14</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">55</span></span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">Total</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">___</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">41</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">___</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><span style=\"color:black\">___</span></span></div>\n			</td>\n		</tr>\n	</tbody>\n</table>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">a. Complete the table.</span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#000000\">Solution&nbsp;3.21</span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">a. There are 45 females in the sample; 18 prefer the coastline and 16 prefer hiking near lakes and streams. So, we know there are 45 &minus; 18 &minus; 16 = 11 female students who prefer hiking on mountain peaks.</span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">Continue reasoning in this way to complete the table.</span></span></p>\n\n<p>&nbsp;</p>\n\n<table cellspacing=\"0\" class=\"Table\" style=\"border-collapse:collapse; border:none; width:900px\">\n	<caption>\n	<p><span style=\"font-size:18px\"><strong>Table3.6 Hiking Area Preference</strong></span></p>\n	</caption>\n	<thead>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Sex</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>The Coastline</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Near Lakes and Streams</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>On Mountain Peaks</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Total</strong></span></div>\n			</td>\n		</tr>\n	</thead>\n	<tbody>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">Female</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">18</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">16</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>11</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">45</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">Male</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>16</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>25</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">14</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">55</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">Total</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>34</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">41</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>25</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>100</strong></span></div>\n			</td>\n		</tr>\n	</tbody>\n</table>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">b. Are the events&nbsp;<em>being female</em>&nbsp;and&nbsp;<em>preferring the coastline</em>&nbsp;independent events?</span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">Let&nbsp;<em>F</em>&nbsp;= being female and let&nbsp;<em>C</em>&nbsp;= preferring the coastline.</span></span></p>\n\n<p>&nbsp;</p>\n\n<ol>\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">Find&nbsp;<em>P</em>(<em>F</em>&nbsp;AND&nbsp;<em>C</em>).</span></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">Find&nbsp;<em>P</em>(<em>F</em>)<em>P</em>(<em>C</em>).</span></span></p>\n	</li>\n</ol>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">Are these two numbers the same? If they are, then&nbsp;<em>F</em>&nbsp;and&nbsp;<em>C</em>&nbsp;are independent. If they are not, then&nbsp;<em>F</em>&nbsp;and&nbsp;<em>C</em>&nbsp;are not independent.</span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution3.&nbsp;21</strong></span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#424242\">b.</span></span></p>\n\n<ol>\n	<li>\n	<p><span style=\"font-size:14px\"><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(<em>F</em>&nbsp;AND&nbsp;<em>C</em>)&nbsp;<span class=\"math-tex\">\\(=\\frac{18}{100}=.18\\)</span> </span></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(<em>F</em>)<em>P</em>(<em>C</em>)&nbsp;<span class=\"math-tex\">\\(=(\\frac{45}{100})(\\frac{34}{100})=(.45)(.34)=.153\\)</span></span></span></p>\n	</li>\n</ol>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(<em>F</em>&nbsp;AND&nbsp;<em>C</em>) &ne;&nbsp;<em>P</em>(<em>F</em>)<em>P</em>(<em>C</em>), so the events&nbsp;<em>F</em>&nbsp;and&nbsp;<em>C</em>&nbsp;are not independent.</span></span><br />\n<br />\n&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#424242\">c</span><span style=\"color:#002060\">. </span><span style=\"color:black\">Find the probability that a person is male given that the person prefers hiking near lakes and streams. Let&nbsp;<em>M</em>&nbsp;= being male, and let&nbsp;<em>L</em>&nbsp;= prefers hiking near lakes and streams.</span></span></p>\n\n<p>&nbsp;</p>\n\n<ol>\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">What word tells you this is a conditional?</span></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">Is the sample space for this problem all 100 hikers? If not, what is it?</span></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">Fill in the blanks and calculate the probability:&nbsp;<em>P</em>(_____|_____) = _____.</span></span></p>\n	</li>\n</ol>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><strong><span style=\"color:#333333\">Solution&nbsp;3</span></strong></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#424242\">c.</span></span></p>\n\n<ol>\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">The word&nbsp;<em>given</em>&nbsp;tells you that this is a conditional.</span></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">No, the sample space for this problem is the 41 hikers who prefer lakes and streams.</span></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">Find the conditional probability&nbsp;<em>P</em>(<em>M</em>|<em>L</em>). Because it is given that the person prefers hiking near lakes and streams, you need only consider the values in the column labeled &quot;Near Lakes and Streams.&quot;&nbsp;<em>P</em>(<em>M</em>|<em>L</em>)<span class=\"math-tex\">\\(=\\frac{25}{41}\\)</span> </span></span></p>\n	</li>\n</ol>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#424242\">d. </span><span style=\"color:black\">Find the probability that a person is female or prefers hiking on mountain peaks. Let&nbsp;<em>F</em>&nbsp;= being female, and let&nbsp;<em>P</em>&nbsp;= prefers mountain peaks.</span></span></p>\n\n<ol>\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">Find&nbsp;<em>P</em>(<em>F</em>).</span></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">Find&nbsp;<em>P</em>(<em>P</em>).</span></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">Find&nbsp;<em>P</em>(<em>F</em>&nbsp;AND&nbsp;<em>P</em>).</span></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:black\">Find&nbsp;<em>P</em>(<em>F</em>&nbsp;OR&nbsp;<em>P</em>).</span></span></p>\n	</li>\n</ol>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><strong><span style=\"color:#333333\">Solution&nbsp;d</span></strong></span></p>\n\n<p>&nbsp;</p>\n\n<ol>\n	<li>\n	<p><span style=\"font-size:14px\"><strong><span style=\"color:#333333\"><span class=\"math-tex\">\\(P(F)=\\frac{45}{100}\\)</span></span></strong></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><strong><span style=\"color:#333333\"><span class=\"math-tex\">\\(P(P)=\\frac{25}{100}\\)</span></span></strong></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><strong><span style=\"color:#333333\"><span class=\"math-tex\">\\(P(F\\hspace{0.1cm}AND\\hspace{0.1cm}P)=\\frac{number\\hspace{0.1cm}of\\hspace{0.1cm}hikers\\hspace{0.1cm}that\\hspace{0.1cm}are\\hspace{0.1cm}both\\hspace{0.1cm}female\\hspace{0.1cm}AND\\hspace{0.1cm}prefers\\hspace{0.1cm}mountain\\hspace{0.1cm}peaks}{number\\hspace{0.1cm}of\\hspace{0.1cm}hikers\\hspace{0.1cm}in\\hspace{0.1cm}study}=\\frac{11}{100}\\)</span>&nbsp;</span></strong></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(<em>F</em>&nbsp;OR&nbsp;<em>P</em>) =&nbsp;<em>P</em>(<em>F</em>) +&nbsp;<em>P</em>(<em>P</em>) &minus;&nbsp;<em>P</em>(<em>F</em>&nbsp;AND&nbsp;<em>P</em>) </span>=<span class=\"math-tex\">\\(\\frac{45}{100}+\\frac{25}[100}-\\frac{11}{100}=\\frac{59}{100}\\)</span></span></p>\n	</li>\n</ol>\n\n<p>&nbsp;</p>\n\n<p><strong><span style=\"font-size:14px\"><span style=\"color:#002060\">EXAMPLE&nbsp;3.22</span></span></strong></p>\n\n<p>&nbsp;</p>\n\n<p style=\"text-align:justify\"><span style=\"font-size:14px\">Muddy Mouse lives in a cage with three doors. If Muddy goes out the first door, the probability that he gets caught by Alissa the cat is&nbsp;1515&nbsp;and the probability he is not caught is&nbsp;4545. If he goes out the second door, the probability he gets caught by Alissa is&nbsp;1414&nbsp;and the probability he is not caught is&nbsp;3434. The probability that Alissa catches Muddy coming out of the third door is&nbsp;1212&nbsp;and the probability she does not catch Muddy is&nbsp;1212. It is equally likely that Muddy will choose any of the three doors, so the probability of choosing each door is&nbsp;1313.</span></p>\n\n<p style=\"text-align:justify\">&nbsp;</p>\n\n<table cellspacing=\"0\" class=\"Table\" style=\"border-collapse:collapse; border:none; width:900px\">\n	<caption>\n	<p><span style=\"font-size:18px\"><strong>Table3.8 Door Choice</strong></span></p>\n	</caption>\n	<thead>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Caught or Not</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Door One</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Door Two</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Door Three</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Total</strong></span></div>\n			</td>\n		</tr>\n	</thead>\n	<tbody>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">Caught</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center\">\n			<div><span style=\"font-size:28px\"><span class=\"math-tex\">\\(\\frac{1}{15}\\)</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center\">\n			<div><span style=\"font-size:28px\"><span class=\"math-tex\">\\(\\frac{1}{12}\\)</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center\">\n			<div><span style=\"font-size:28px\"><span class=\"math-tex\">\\(\\frac{1}{6}\\)</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center\">\n			<div>&nbsp;</div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">Not Caught</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center; vertical-align:top\">\n			<div><span style=\"font-size:28px\"><span class=\"math-tex\">\\(\\frac{4}{15}\\)</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center; vertical-align:top\">\n			<div><span style=\"font-size:28px\"><span class=\"math-tex\">\\(\\frac{3}{12}\\)</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center; vertical-align:top\">\n			<div><span style=\"font-size:28px\"><span class=\"math-tex\">\\(\\frac{1}{6}\\)</span></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center\">\n			<div>&nbsp;</div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">Total</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">____</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">____</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">____</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">1</span></div>\n			</td>\n		</tr>\n	</tbody>\n</table>\n\n<p>&nbsp;</p>\n\n<ul>\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:#424242\">The first entry&nbsp;<span class=\"math-tex\">\\(\\frac{1}{15}=(\\frac{1}{5})(\\frac{1}{3})\\)</span></span>&nbsp;<em><span style=\"color:#424242\">P</span></em><span style=\"color:#424242\">(Door One AND Caught).</span></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:#424242\">The entry&nbsp;</span><span class=\"math-tex\">\\(\\frac{1}{15}=(\\frac{!}{5})(\\frac{1}{3})\\)</span>&nbsp;<span style=\"color:#424242\"><em>P</em>(Door One AND Not Caught).</span></span></p>\n	</li>\n</ul>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#000000\">Verify the remaining entries.</span></span></p>\n\n<p><br />\n<br />\n<span style=\"font-size:14px\"><span style=\"color:#002060\">a. </span><span style=\"color:black\">Complete the probability contingency table. Calculate the entries for the totals. Verify that the lower-right corner entry is 1.</span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution&nbsp;3.22</strong></span></span></p>\n\n<table cellspacing=\"0\" class=\"Table\" style=\"border-collapse:collapse; border:none; width:900px\">\n	<caption>\n	<p><span style=\"font-size:18px\"><strong>Table3.9 Door Choice</strong></span></p>\n	</caption>\n	<thead>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Caught or Not</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Door One</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Door Two</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Door Three</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Total</strong></span></div>\n			</td>\n		</tr>\n	</thead>\n	<tbody>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">Caught</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">115115</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">112112</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">1616</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">19601960</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">Not Caught</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">415415</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">312312</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">1616</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">41604160</span></div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">Total</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">515515</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">412412</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">2626</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">1</span></div>\n			</td>\n		</tr>\n	</tbody>\n</table>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#424242\">b</span><span style=\"color:black\">. What is the probability that Alissa does not catch Muddy?</span></span></p>\n\n<h4>&nbsp;</h4>\n\n<p><strong><span style=\"font-size:14px\"><span style=\"color:#2e74b5\"><span style=\"color:black\">Solution&nbsp;b</span></span></span></strong></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:black\">b.&nbsp;41604160</span></span></p>\n\n<p><br />\n<br />\n<span style=\"font-size:14px\"><span style=\"color:black\">c. What is the probability that Muddy chooses Door One OR Door Two given that Muddy is caught by Alissa?</span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><strong><span style=\"color:#2e74b5\"><span style=\"color:#333333\">Solution&nbsp;c</span></span></strong></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#424242\">c. This is a conditional probability, so consider only probabilities in the row labeled &quot;Caught.&quot; Choosing Door One and choosing Door Two are mutually exclusive, so</span>&nbsp;P(Choosing Door One OR Choosing Door Two AND Caught)<span class=\"math-tex\">\\(= \\frac{1}{15}+\\frac{1}{12}=\\frac{9}{60}\\)</span>.</span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#424242\">Use the formula for conditional probability&nbsp;P(A??B)&nbsp;<span class=\"math-tex\">\\(=\\frac{P(A\\hspace{0.1cm}AND\\hspace{0.1cm}B)}{P(B)}\\)</span></span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\">P(Door One OR Door Two|Caught)&nbsp;<span class=\"math-tex\">\\(=\\frac{P(DOOR\\hspace{0.1cm}one\\hspace{0.1cm}OR\\hspace{0.1cm}DOOR\\hspace{0.1cm}TWO\\hspace{0.1cm}AND\\hspace{0.1cm}Caught)}{P(Caught)}=\\frac{\\frac{9}{60}}{\\frac{19}{60}}=\\frac{9}{19}\\)</span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>EXAMPLE&nbsp;3.23</strong></span></span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><a href=\"https://openstax.org/books/statistics/pages/3-4-contingency-tables#Ch03_M04_tbl007\" style=\"color:blue; text-decoration:underline\"><span style=\"color:#027eb5\">Table 3.10</span></a><span style=\"color:#424242\">&nbsp;contains the number of crimes per 100,000 inhabitants from 2008 to 2011 in the United States.\\</span></span></p>\n\n<p>&nbsp;</p>\n\n<table cellspacing=\"0\" class=\"Table\" style=\"border-collapse:collapse; border:none; width:900px\">\n	<thead>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Year</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Crime A</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Crime B</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Crime C</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Crime D</strong></span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:1px solid black; vertical-align:bottom\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\"><strong>Total</strong></span></div>\n			</td>\n		</tr>\n	</thead>\n	<tbody>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">2008</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">145.7</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">732.1</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">29.7</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">314.7</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center\">\n			<div>&nbsp;</div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">2009</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">133.1</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">717.7</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">29.1</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">259.2</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center\">\n			<div>&nbsp;</div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">2010</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">119.3</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">701</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">27.7</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">239.1</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center\">\n			<div>&nbsp;</div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">2011</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">113.7</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">702.2</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">26.8</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">229.6</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center\">\n			<div>&nbsp;</div>\n			</td>\n		</tr>\n		<tr>\n			<td style=\"border-bottom:1px solid black; border-left:1px solid black; border-right:1px solid black; border-top:none\">\n			<div style=\"text-align:center\"><span style=\"font-size:28px\">Total</span></div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center\">\n			<div>&nbsp;</div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center\">\n			<div>&nbsp;</div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center\">\n			<div>&nbsp;</div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center\">\n			<div>&nbsp;</div>\n			</td>\n			<td style=\"border-bottom:1px solid black; border-left:none; border-right:1px solid black; border-top:none; text-align:center\">\n			<div>&nbsp;</div>\n			</td>\n		</tr>\n	</tbody>\n</table>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><strong>Table3.10</strong>&nbsp;U.S. Crime Index Rates Per 100,000 Inhabitants 2008&ndash;2011</span></p>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#424242\">TOTAL each column and each row. Total data = 4,520.7.</span></span></p>\n\n<p>&nbsp;</p>\n\n<ol style=\"list-style-type:lower-alpha\">\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:#424242\">Find&nbsp;<em>P</em>(2009 AND Crime A).</span></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:#424242\">Find&nbsp;<em>P</em>(2010 AND Crime B).</span></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:#424242\">Find&nbsp;<em>P</em>(2010 OR Crime B).</span></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:#424242\">Find&nbsp;<em>P</em>(2011|Crime A).</span></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><span style=\"color:#424242\">Find&nbsp;<em>P</em>(Crime D|2008).</span></span></p>\n	</li>\n</ol>\n\n<p>&nbsp;</p>\n\n<p><span style=\"font-size:14px\"><span style=\"color:#000000\"><strong>Solution&nbsp;3.23</strong></span></span></p>\n\n<p>&nbsp;</p>\n\n<ol start=\"1\" style=\"list-style-type:lower-alpha\">\n	<li>\n	<p><span style=\"font-size:14px\"><strong><span style=\"color:#002060\"><span class=\"math-tex\">\\(\\frac{133.14}{520.7}=.0294\\)</span></span></strong></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><span class=\"math-tex\">\\(\\frac{7014}{520.7}=.1551\\)</span></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><em><span style=\"color:black\">P</span></em><span style=\"color:black\">(2010 OR Crime B) =&nbsp;<em>P</em>(2010) +&nbsp;<em>P</em>(Crime B) &ndash;&nbsp;<em>P</em>(2010 AND Crime B) <span class=\"math-tex\">\\(\\frac{1,087.14}{520.7}+\\frac{2,852.94}{520.7}-\\frac{7014}{520.7}=.7165\\)</span></span>\\</span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><span class=\"math-tex\">\\(\\frac{113.7}{511.8}=.2222\\)</span></span></p>\n	</li>\n	<li>\n	<p><span style=\"font-size:14px\"><span class=\"math-tex\">\\(\\frac{314.7}{1,222.2}=.2575\\)</span></span></p>\n	</li>\n</ol>','','1','3.4',NULL,'1','2','1','0','1','2021-06-23 06:22:36','2021-06-23 11:52:36'),(2269,97,'3.5 Tree and Venn Diagrams','','single','html','<p style=\"text-align:justify\"><span style=\"font-size:14px\"><span style=\"color:black\">Sometimes, when the probability problems are complex, it can be helpful to graph the situation. Tree diagrams and Venn diagrams are two tools that can be used to visualize and solve conditional probabilities</span><span style=\"color:#424242\">.</span></span></p>','','1','3.5',NULL,'1','2','1','0','1','2021-06-23 06:22:58','2021-06-23 11:52:58'),