To solve this problem, let's find the five-number summary for the given set of test scores and then outline the process for drawing a box-and-whisker plot based on that summary.

Step 1: Organize the Data

The data given is already sorted in ascending order: $31, 41, 45, 48, 52, 55, 56, 63, 65, 67, 67, 69, 70, 74, 75, 78, 79, 79, 80, 83, 85, 85, 87, 90, 92, 95$

Step 2: Calculate the Five-Number Summary

The five-number summary consists of:

1. Minimum: The smallest value in the data set.
2. First Quartile (Q1): The median of the lower half of the data (below the median).
3. Median (Q2): The middle value of the entire data set.
4. Third Quartile (Q3): The median of the upper half of the data (above the median).
5. Maximum: The largest value in the data set.

Minimum and Maximum

• Minimum: 31
• Maximum: 95

Median (Q2)

Since there are 28 values, the median will be the average of the 14th and 15th values: $\text{Median} = \frac{70 + 74}{2} = 72$

First Quartile (Q1)

Q1 is the median of the lower 14 values: $31, 41, 45, 48, 52, 55, 56, 63, 65, 67, 67, 69, 70$ The 7th and 8th values in this subset are 56 and 63. $Q1 = \frac{56 + 63}{2} = 59.5$

Third Quartile (Q3)

Q3 is the median of the upper 14 values: $74, 75, 78, 79, 79, 80, 83, 85, 85, 87, 90, 92, 95$ The 7th and 8th values in this subset are 83 and 85. $Q3 = \frac{83 + 85}{2} = 84$

Summary

The five-number summary is:

• Minimum: 31
• Q1: 59.5
• Median (Q2): 72
• Q3: 84
• Maximum: 95

Step 3: Drawing the Box-and-Whisker Plot

To draw the box-and-whisker plot:

1. Draw a number line that includes the minimum and maximum values.
2. Mark the five-number summary points (Minimum, Q1, Median, Q3, Maximum) on the number line.
3. Draw a rectangular "box" from Q1 to Q3, with a line inside the box at the Median.
4. Extend "whiskers" from the box out to the Minimum and Maximum values.

Would you like further clarification or a more detailed guide on how to sketch the box plot?

Here are some additional questions to consider:

1. How would you calculate the interquartile range (IQR) for this data?
2. What could outliers look like in a box-and-whisker plot, and how are they determined?
3. How can you interpret the spread of data in terms of quartiles?
4. What does a larger spread between Q1 and Q3 indicate about the dataset?
5. How does this five-number summary help in understanding the dataset's distribution?

Tip: Always double-check each quartile calculation, as accuracy is crucial in interpreting the box plot.