<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>Example 2.20</strong></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<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>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>Solution 2.20</strong></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<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>

<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>

<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>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>Example 2.21</strong></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<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>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>Solution 2.21</strong></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">&bull; Seventy percent of students answered 16 or fewer questions correctly.</span></p>

<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>

<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>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>Example 2.22</strong></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<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>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>Solution 2.22</strong></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<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>

<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>

<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>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>Example 2.23</strong></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<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>

<p style="text-align:justify">&nbsp;</p>

<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>

<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>

<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>

<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>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Min = 0</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Q1 = 20</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Med = 40</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Q3 = 60</span></span></p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Max = 300</span></span></p>

<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>

<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>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:#000000"><strong>Figure 2.12</strong></span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">The minimum value is 0.</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">The maximum value is 300.</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<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>

<p style="text-align:justify">&nbsp;</p>

<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>

<p style="text-align:justify">&nbsp;</p>

<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>

<p style="text-align:justify">&nbsp;</p>

<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>

<p style="text-align:justify">&nbsp;</p>

<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>

<p style="text-align:justify">&nbsp;</p>

<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>

<p style="text-align:justify">&nbsp;</p>

<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>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><span style="color:black">Min = 0</span></span></p>

<p style="text-align:justify">&nbsp;</p>

<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>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px"><em>Q</em>3 = 60</span></p>

<p style="text-align:justify">&nbsp;</p>

<p style="text-align:justify"><span style="font-size:14px">Max = 120</span></p>

<p style="text-align:justify">&nbsp;</p>

<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>