Question

the following are the ages of 18 history teachers in a school district. 24, 25, 26, 29, 29, 31, 33, 37, 42, 44, 45, 46, 47, 49, 49, 52, 56, 57. notice that the ages are ordered from least to greatest. make a box - and - whisker plot for the data.

Explanation:

Step1: Find minimum and maximum

The minimum is \(24\) and maximum is \(57\) from the data set.

Step2: Calculate median

For \(n = 18\) (even), \(Q_2=\frac{42 + 44}{2}=43\).

Step3: Calculate first quartile

For lower half (\(n_1 = 9\) odd), \(Q_1\) is 5th value of lower - half data, \(Q_1 = 29\).

Step4: Calculate third quartile

For upper half (\(n_2 = 9\) odd), \(Q_3\) is 5th value of upper - half data, \(Q_3=49\).

Step5: Draw the plot

Place dots for min and max, draw box from \(Q_1\) to \(Q_3\) and vertical line at \(Q_2\).

Answer:

To create a box - and - whisker plot, we need to find the five - number summary: minimum, first quartile ($Q_1$), median ($Q_2$), third quartile ($Q_3$), and maximum.

1. Minimum and Maximum:

• The data set is \(24,25,26,29,29,31,33,37,42,44,45,46,47,49,49,52,56,57\).
• The minimum value is \(24\) and the maximum value is \(57\).

1. Median ($Q_2$):

• Since \(n = 18\) (an even number), the median is the average of the \(\frac{n}{2}\)th and \((\frac{n}{2}+ 1)\)th ordered data values.
• \(\frac{n}{2}=\frac{18}{2}=9\) and \(\frac{n}{2}+1 = 10\).
• The 9th value is \(42\) and the 10th value is \(44\). So, \(Q_2=\frac{42 + 44}{2}=43\).

1. First Quartile ($Q_1$):

• Consider the lower half of the data (the first 9 values: \(24,25,26,29,29,31,33,37,42\)).
• Since \(n_1 = 9\) (an odd number), the first - quartile is the \(\frac{n_1 + 1}{2}\)th value.
• \(\frac{9+1}{2}=5\)th value. So, \(Q_1 = 29\).

1. Third Quartile ($Q_3$):

• Consider the upper half of the data (the last 9 values: \(44,45,46,47,49,49,52,56,57\)).
• Since \(n_2=9\) (an odd number), the third - quartile is the \(\frac{n_2 + 1}{2}\)th value.
• \(\frac{9 + 1}{2}=5\)th value. So, \(Q_3=49\).

On the number line:

• Place a dot at the minimum value \(24\) (left - most whisker end).
• Place a dot at the maximum value \(57\) (right - most whisker end).
• Draw a box from \(Q_1 = 29\) to \(Q_3=49\).
• Draw a vertical line inside the box at the median \(Q_2 = 43\).