(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…

(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.)

(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.)

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).