Question

a modified box-and-whisker plot is a box-and-whisker plot that uses symbols to identify outliers. the horizontal line of a modified box-and-whisker plot extends as far as the minimum data entry that is not an outlier and the maximum data entry that is not an outlier. (a) identify any outliers and (b) draw a modified box-and-whisker plot that represents the data set. use open circles (∘) to plot any outliers. 26 43 48 61 57 52 18 94 3 49 48 64 52 55 69 39 53 31 (a) identify any possible outliers. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the outlier(s) is(are) (type an integer or decimal. use a comma to separate answers as needed.) b. there are no outliers.

Explanation:

Step 1: Order the data

First, we order the data set from smallest to largest: \(3, 18, 26, 31, 39, 43, 48, 48, 49, 52, 52, 53, 55, 57, 61, 64, 69, 94\)

Step 2: Find the quartiles

• The median (second quartile, \(Q_2\)) of the data set with \(n = 18\) (even number of data points) is the average of the 9th and 10th values. The 9th value is \(49\) and the 10th value is \(52\), so \(Q_2=\frac{49 + 52}{2}=50.5\)
• The first quartile (\(Q_1\)) is the median of the lower half of the data (the first 9 values: \(3, 18, 26, 31, 39, 43, 48, 48, 49\)). The median of these 9 values (odd number) is the 5th value, so \(Q_1 = 39\)
• The third quartile (\(Q_3\)) is the median of the upper half of the data (the last 9 values: \(52, 52, 53, 55, 57, 61, 64, 69, 94\)). The median of these 9 values (odd number) is the 5th value, so \(Q_3=57\)

Step 3: Calculate the interquartile range (IQR)

The interquartile range is \(IQR=Q_3 - Q_1\). So \(IQR = 57-39 = 18\)

Step 4: Determine the outlier boundaries

• Lower bound: \(Q_1-1.5\times IQR=39 - 1.5\times18=39 - 27 = 12\)
• Upper bound: \(Q_3 + 1.5\times IQR=57+1.5\times18=57 + 27 = 84\)

Step 5: Identify outliers

We check each data point:

• Values less than 12: \(3\) (since \(3<12\))
• Values greater than 84: \(94\) (since \(94 > 84\))

Answer:

3, 94