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Chapter 3: Problem 99

Prepare a box-and-whisker plot for the following data: \(\begin{array}{llllllll}36 & 43 & 28 & 52 & 41 & 59 & 47 & 61 \\ 24 & 55 & 63 & 73 & 32 & 25 & 35 & 49 \\ 31 & 22 & 61 & 42 & 58 & 65 & 98 & 34\end{array}\) Does this data set contain any outliers?

Short Answer

Expert verified
The box-and-whisker plot can be made for the provided data set using the identified five number summary. The data set does not contain any outliers.

Step by step solution

01

Organize the Data

First sort the data from smallest to largest. This becomes 22, 24, 25, 28, 31, 32, 34, 35, 36, 41, 42, 43, 47, 49, 52, 55, 58, 59, 61, 61, 63, 65, 73, 98.
02

Determine the Five Number Summary

Next, find the smallest number (22), lower quartile \(Q1 = 32.5\) (the median of the lower half of the data), median \(Q2 = 43.5\), upper quartile \(Q3 = 61\) (the median of the upper half of the data), and the highest number (98).
03

Make the Box-Whisker Plot

Start by putting a scale on a number line that includes the smallest and largest numbers in the data set. Then draw a box that starts at Q1 and ends at Q3. Draw an additional line (the 'whisker') from the smallest number to Q1, and another line from Q3 to the largest number. Put a line in the box at Q2 to represent the median.
04

Determine if there are outliers

Find the inner quartile range (IQR), which equals Q3 - Q1 = 61 - 32.5 = 28.5. To identify outliers, multiply IQR by 1.5 and then subtract this value from Q1 (32.5 - 1.5*28.5 = -10) and add it to Q3 (61 + 1.5*28.5 = 103.75). Therefore, anything below -10 and above 103.75 would be considered outliers. In this case, the data set does not contain any outliers because all the numbers fall within this range.

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