the following are the distances (in miles) to the nearest airport for 18 families. 8, 14, 15, 20, 21, 22…

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 minimum value in the data set is $8$.
   • The maximum value in the data set is $49$.
2. Median ($Q_2$):
   • Since there are $n = 18$ data points, the median is the average of the $\frac{n}{2}=9$th and $(\frac{n}{2}+1) = 10$th ordered data points.
   • The 9th value is $26$ and the 10th value is $29$. So, $Q_2=\frac{26 + 29}{2}=27.5$.
3. First Quartile ($Q_1$):
   • The lower half of the data consists of the first 9 data points: $8,14,15,20,21,22,23,24,26$.
   • Since there are 9 data points in the lower half, the first quartile is the 5th value. So, $Q_1 = 21$.
4. Third Quartile ($Q_3$):
   • The upper half of the data consists of the last 9 data points: $29,31,33,39,45,47,48,49$.
   • Since there are 9 data points in the upper half, the third quartile is the 5th value of the upper - half data. So, $Q_3=39$.

On the box - and - whisker plot:

• The left - most point (the end of the left whisker) is at $8$ (the minimum).
• The left - hand side of the box is at $Q_1 = 21$.
• The line inside the box is at $Q_2=27.5$.
• The right - hand side of the box is at $Q_3 = 39$.
• The right - most point (the end of the right whisker) is at $49$ (the maximum).

Explanation:

Step1: Identify minimum and maximum

Minimum = $8$, Maximum = $49$

Step2: Calculate median

$n = 18$, $Q_2=\frac{26 + 29}{2}=27.5$

Step3: Find first quartile

Lower half has 9 points, $Q_1 = 21$

Step4: Find third quartile

Upper half has 9 points, $Q_3=39$