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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, we order the data set from smallest to largest: $20,25,30,35,40,40,60,60,75,90$. The minimum value is $20$.
2. First quartile ($Q_1$):
   • The data set has $n = 10$ values. The position of $Q_1$ is at $\frac{n + 1}{4}=\frac{10+1}{4}=2.75$.
   • Interpolating between the 2nd and 3rd ordered values. The 2nd value is $25$ and the 3rd value is $30$. So $Q_1=25+(30 - 25)\times0.75=25 + 3.75=28.75$.
3. Median:
   • The position of the median for $n = 10$ (an even - numbered data set) is at $\frac{n}{2}=5$ and $\frac{n}{2}+1 = 6$. The median is the average of the 5th and 6th ordered values. So, median$=\frac{40 + 40}{2}=40$.
4. Third quartile ($Q_3$):
   • The position of $Q_3$ is at $\frac{3(n + 1)}{4}=\frac{3\times(10 + 1)}{4}=8.25$.
   • Interpolating between the 8th and 9th ordered values. The 8th value is $60$ and the 9th value is $75$. So $Q_3=60+(75 - 60)\times0.25=60 + 3.75=63.75$.
5. Maximum value:
   • The maximum value in the ordered data set is $90$.

We then look at the box - plots to see which one has a minimum of $20$, $Q_1\approx29$, median of $40$, $Q_3\approx64$, and maximum of $90$.

Explanation:

Step1: Order the data

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

Step2: Find the minimum

The minimum is $20$.

Step3: Calculate $Q_1$

Position is $2.75$, $Q_1 = 28.75$

Step4: Find the median

Average of 5th and 6th values, median$ = 40$

Step5: Calculate $Q_3$

Position is $8.25$, $Q_3=63.75$

Step6: Find the maximum

The maximum is $90$.

Step7: Match with box - plot

Match the five - number summary with the box - plots.