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Answer:

To determine the correct box - plot, we need to find the five - number summary (minimum, first quartile $Q_1$, median, third quartile $Q_3$, maximum) of the data set ${20,25,30,35,40,40,60,60,75,90}$.

1. Minimum value:
   • First, arrange the data in ascending order: $20,25,30,35,40,40,60,60,75,90$. The minimum value is $20$.
2. Median ($Q_2$):
   • Since there are $n = 10$ data points, the median is the average of the $\frac{n}{2}=5$th and $\left(\frac{n}{2}+1\right)=6$th ordered data points. So, $Q_2=\frac{40 + 40}{2}=40$.
3. First quartile ($Q_1$):
   • The lower half of the data is ${20,25,30,35,40}$. Since there are $n_1 = 5$ data points in the lower half, the first quartile is the $\frac{n_1 + 1}{2}=3$rd ordered data point in the lower half. So, $Q_1 = 30$.
4. Third quartile ($Q_3$):
   • The upper half of the data is ${60,60,75,90}$. Since there are $n_2=5$ data points in the upper half, the third quartile is the $\frac{n_2 + 1}{2}=3$rd ordered data point in the upper half. So, $Q_3 = 60$.
5. Maximum value:
   • The maximum value is $90$.

The box - plot should have a left - whisker starting at $20$, the left edge of the box at $30$, the line inside the box at $40$, the right edge of the box at $60$, and the right - whisker ending at $90$.

We need to visually inspect the given box - plots to find the one that matches these values.

Explanation:

Step1: Arrange data in ascending order

$20,25,30,35,40,40,60,60,75,90$

Step2: Calculate minimum

The minimum is $20$ as it is the smallest value in the ordered data.

Step3: Calculate median

$Q_2=\frac{40 + 40}{2}=40$ (average of 5th and 6th values for $n = 10$ data points)

Step4: Calculate first quartile

For lower half ${20,25,30,35,40}$, $Q_1$ is the 3rd value, so $Q_1 = 30$

Step5: Calculate third quartile

For upper half ${60,60,75,90}$, $Q_3$ is the 3rd value, so $Q_3 = 60$

Step6: Calculate maximum

The maximum is $90$ as it is the largest value in the ordered data. Then match with box - plots.