Question

1. find the five number summary of the following data set:

12, 15, 9, 5, 17, 16, 10, 11, 4, 8, 9, 20, 12

1. construct a box and whisker plot for the data set in question 12.

Explanation:

Question 12 Solution:

Step1: Order the data set

First, we need to sort the data set in ascending order. The given data set is: \(12, 15, 9, 5, 17, 16, 10, 11, 4, 8, 9, 20, 12\). After sorting, we get: \(4, 5, 8, 9, 9, 10, 11, 12, 12, 15, 16, 17, 20\).

Step2: Find the minimum value

The minimum value is the smallest number in the sorted data set. From the sorted data, the minimum value is \(4\).

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

The first quartile is the median of the lower half of the data. The number of data points \(n = 13\), which is odd. The lower half of the data (excluding the median) is the first \(\frac{n - 1}{2}=6\) numbers: \(4, 5, 8, 9, 9, 10\). The median of this lower half is the average of the 3rd and 4th numbers (since there are 6 numbers, an even count). The 3rd number is \(8\) and the 4th number is \(9\). So, \(Q_1=\frac{8 + 9}{2}=\frac{17}{2} = 8.5\).

Step4: Find the median (second quartile, \(Q_2\))

The median is the middle number of the sorted data set. Since \(n = 13\), the median is the \(\frac{13 + 1}{2}=7\)th number. From the sorted data, the 7th number is \(11\). So, the median \(Q_2 = 11\).

Step5: Find the third quartile (\(Q_3\))

The third quartile is the median of the upper half of the data. The upper half of the data (excluding the median) is the last \(\frac{n - 1}{2}=6\) numbers: \(12, 12, 15, 16, 17, 20\). The median of this upper half is the average of the 3rd and 4th numbers. The 3rd number is \(15\) and the 4th number is \(16\). So, \(Q_3=\frac{15 + 16}{2}=\frac{31}{2}=15.5\).

Step6: Find the maximum value

The maximum value is the largest number in the sorted data set. From the sorted data, the maximum value is \(20\).

To construct a box - and - whisker plot, we use the five - number summary from question 12: Minimum \(=4\), \(Q_1 = 8.5\), Median \(=11\), \(Q_3=15.5\), Maximum \( = 20\).

1. Draw a number line that covers the range of the data (from \(4\) to \(20\) in this case).
2. Plot the minimum (\(4\)) and maximum (\(20\)) values as the endpoints of the whiskers.
3. Draw a box from \(Q_1 = 8.5\) to \(Q_3=15.5\). Inside the box, draw a vertical line at the median value (\(11\)).

Answer:

The five - number summary is: Minimum \( = 4\), \(Q_1=8.5\), Median (\(Q_2\)) \( = 11\), \(Q_3 = 15.5\), Maximum \( = 20\)

Question 13 Solution: