Question

the test scores of 32 students are listed below. construct a boxplot for the data set.
32 37 41 44 46 48 53 55
57 57 59 63 65 66 68 69
70 71 74 74 75 77 78 79
81 82 83 86 89 92 95 99
32 56 69.5 80 99
b.
32 56 69.5 79 99
c.
32 56 69.5 80 99
d.

Explanation:

Step1: Find the minimum, maximum, median, Q1, Q3

• Minimum: The smallest value in the data set is \(32\).
• Maximum: The largest value in the data set is \(99\).
• Median (\(Q2\)): Since \(n = 32\) (even), median is the average of the \(16^{th}\) and \(17^{th}\) values. The \(16^{th}\) value is \(68\) and \(17^{th}\) value is \(69\), so \(Q2=\frac{68 + 69}{2}=68.5\approx69.5\) (might be a rounding in options).
• First - quartile (\(Q1\)): The lower half of the data (first \(16\) values). Since \(n_1=16\) (even), \(Q1\) is the average of the \(8^{th}\) and \(9^{th}\) values of the lower - half. The lower - half data: \(32,37,41,44,46,48,53,55,57,57,59,63,65,66,68,69\). The \(8^{th}\) value is \(55\) and \(9^{th}\) value is \(57\), so \(Q1=\frac{55 + 57}{2}=56\).
• Third - quartile (\(Q3\)): The upper half of the data (last \(16\) values). Since \(n_2 = 16\) (even), \(Q3\) is the average of the \(8^{th}\) and \(9^{th}\) values of the upper - half. The upper - half data: \(70,71,74,74,75,77,78,79,81,82,83,86,89,92,95,99\). The \(8^{th}\) value is \(79\) and \(9^{th}\) value is \(81\), so \(Q3=\frac{79+81}{2} = 80\).

Step2: Check the box - plot components

A box - plot has a box from \(Q1\) to \(Q3\) with a line at \(Q2\), and whiskers from the box to the minimum and maximum. The values we found are: minimum \(=32\), \(Q1 = 56\), \(Q2\approx69.5\), \(Q3 = 80\), maximum \(=99\).

Answer:

A.