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Chapter 3: Problem 100

Prepare a box-and-whisker plot for the following data: \(\begin{array}{lrrrrrrrrr}11 & 8 & 26 & 31 & 62 & 19 & 7 & 3 & 14 & 75 \\ 33 & 30 & 42 & 15 & 18 & 23 & 29 & 13 & 16 & 6\end{array}\) Does this data set contain any outliers?

Short Answer

Expert verified
In this data set, the outliers are 62 and 75.

Step by step solution

01

Arrange the data

First, arrange the given data in ascending order to determine the lowest and highest values, as well as the quartiles. The arranged data is: 3, 6, 7, 8, 11, 13, 14, 15, 16, 18, 19, 23, 26, 29, 30, 31, 33, 42, 62, 75.
02

Determine Quartiles

Once the data is arranged in increasing order, the quartiles can be calculated. The median, or second quartile (Q2), is the average of the 10th and 11th values, which is \((18+19)/2 = 18.5\). Similarly, the first quartile (Q1) is the median of the first half of the data, which is the average of the 5th and 6th values, \((11+13)/2 = 12\). Lastly, the third quartile (Q3) is the median of the second half of the data, which is the average of the 15th and 16th values \((30+31)/2 = 30.5\). The minimum value is 3 and the maximum value is 75.
03

Determine Interquartile Range

The interquartile range (IQR) can be calculated by subtracting the first quartile (Q1) from the third quartile (Q3). The IQR is \(30.5 - 12 = 18.5\).
04

Identify Outliers

Outliers are values that are 1.5 times the IQR below the first quartile or above the third quartile. Calculating this gives \(Q1 - 1.5*IQR = 12 - 1.5*18.5 = -15.75\) and \(Q3 + 1.5*IQR = 30.5 + 1.5*18.5 = 58.25\). Thus, any data points below -15.75 or above 58.25 are considered outliers. Here, the outliers are 62 and 75.

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