Question

(a) find the five - number summary, and (b) draw a box - and - whisker plot of the data. 3 8 8 6 2 9 8 7 9 6 9 5 2 6 2 9 8 7 7 9 (a) min = (simplify your answer.)

Explanation:

Step1: Sort the data

First, sort the data set \(3,8,8,6,2,9,8,7,9,6,9,5,2,6,2,9,8,7,7,9\) in ascending order: \(2,2,2,3,5,6,6,6,7,7,7,8,8,8,8,9,9,9,9,9\).

Step2: Find the minimum

The minimum value of the sorted - data set is the first value. So, \(Min = 2\).

Step3: Find the first quartile \(Q_1\)

The data set has \(n = 20\) values. The position of \(Q_1\) is \(\frac{n + 1}{4}=\frac{20+1}{4}=5.25\). The first quartile is the value between the 5th and 6th ordered values. The 5th value is \(5\) and the 6th value is \(6\), so \(Q_1=5+(6 - 5)\times0.25 = 5.25\).

Step4: Find the median \(Q_2\)

The position of the median for \(n = 20\) (an even - numbered data set) is \(\frac{n}{2}=10\) and \(\frac{n}{2}+1 = 11\). The median \(Q_2=\frac{7 + 7}{2}=7\).

Step5: Find the third quartile \(Q_3\)

The position of \(Q_3\) is \(\frac{3(n + 1)}{4}=\frac{3\times(20 + 1)}{4}=15.75\). The third quartile is the value between the 15th and 16th ordered values. The 15th value is \(8\) and the 16th value is \(9\), so \(Q_3=8+(9 - 8)\times0.75 = 8.75\).

Step6: Find the maximum

The maximum value of the sorted - data set is the last value. So, \(Max = 9\).
The five - number summary is \(Min = 2\), \(Q_1=5.25\), \(Q_2 = 7\), \(Q_3=8.75\), \(Max = 9\).

To draw a box - and - whisker plot:

1. Draw a number line that includes the range from \(2\) to \(9\).
2. Mark the minimum (\(2\)) with a dot or a small circle.
3. Mark the first quartile (\(5.25\)) with a dot or a small circle.
4. Mark the median (\(7\)) with a dot or a small circle.
5. Mark the third quartile (\(8.75\)) with a dot or a small circle.
6. Mark the maximum (\(9\)) with a dot or a small circle.
7. Draw a box from \(Q_1\) to \(Q_3\).
8. Draw a vertical line inside the box at the median.
9. Draw whiskers from the box to the minimum and maximum values.

Answer:

(a) \(Min = 2\), \(Q_1=5.25\), \(Q_2 = 7\), \(Q_3=8.75\), \(Max = 9\)
(b) A box - and - whisker plot is drawn as described above with a number line, a box from \(5.25\) to \(8.75\), a vertical line at \(7\) inside the box, and whiskers extending to \(2\) and \(9\).