Question

which of the data sets below have the following five - number summary? select all that apply.16 - 19 - 23 - 31 - 32a 16, 18, 19, 23, 23, 30, 32, 32b 16, 18, 19, 23, 29, 31, 32c 16, 19, 22, 24, 31, 32d 16, 18, 20, 20, 23, 26, 30, 32, 32

Explanation:

Step1: Find minimum and maximum

The five - number summary is \(16 - 19 - 23 - 31 - 32\). The minimum is \(16\) and the maximum is \(32\) for all options.

Step2: Find median (\(Q_2\))

• For option A: There are \(n = 8\) data points. The median is the average of the \(4^{th}\) and \(5^{th}\) values. \(\frac{23 + 23}{2}=23\).
• For option B: There are \(n = 7\) data points. The median is the \(4^{th}\) value, which is \(23\).
• For option C: There are \(n = 6\) data points. The median is the average of the \(3^{rd}\) and \(4^{th}\) values. \(\frac{22+24}{2} = 23\).
• For option D: There are \(n = 9\) data points. The median is the \(5^{th}\) value, which is \(23\).

Step3: Find \(Q_1\) (first quartile)

• For option A: The lower half is \(16,18,19,23\). There are \(n = 4\) data points. \(Q_1=\frac{18 + 19}{2}=18.5

eq19\).

• For option B: The lower half is \(16,18,19\). There are \(n = 3\) data points. \(Q_1 = 19\).
• For option C: The lower half is \(16,19\). There are \(n = 2\) data points. \(Q_1=\frac{16 + 19}{2}=17.5

eq19\).

• For option D: The lower half is \(16,18,20,20\). There are \(n = 4\) data points. \(Q_1=\frac{18+20}{2} = 19\).

Step4: Find \(Q_3\) (third quartile)

• For option B: The upper half is \(29,31,32\). \(Q_3=31\).
• For option D: The upper half is \(26,30,32,32\). \(Q_3=\frac{30 + 32}{2}=31\).

Answer:

B. \(16, 18, 19, 23, 29, 31, 32\); D. \(16, 18, 20, 20, 23, 26, 30, 32, 32\)