To solve this problem, let’s first organize the five-number summary provided:

• Minimum: 3
• Q1 (First Quartile): 8
• Median (Q2): 10
• Q3 (Third Quartile): 20
• Maximum: 38

(a) What quarter has the smallest spread of data?

• Q1 to Q2: 8 to 10 → Spread = 10 - 8 = 2
• Q2 to Q3: 10 to 20 → Spread = 20 - 10 = 10
• Q3 to Max: 20 to 38 → Spread = 38 - 20 = 18
• Min to Q1: 3 to 8 → Spread = 8 - 3 = 5

The smallest spread is between Q1 and Q2 (Second quarter).
Answer: Second quarter

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(b) What is that spread?

The spread of the second quarter (Q1 to Q2) is 2.
Answer: 2

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(c) What quarter has the largest spread of data?

From the calculations above:

• Q3 to Max (Third quarter) has the largest spread, which is 18.

Answer: Third quarter

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(d) What is that spread?

The spread of the third quarter is 18.
Answer: 18

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(e) Find the Interquartile Range (IQR):

The IQR is the difference between Q3 and Q1: $IQR = Q3 - Q1 = 20 - 8 = 12$
Answer: 12

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(f) Which interval has the most data in it?

• 12 - 36: Not aligned with quartiles.
• 20 - 38: Matches Q3 to Max (contains 25% of the data).
• 10 - 20: Matches Q2 to Q3 (contains 25% of the data).
• 3 - 10: Matches Min to Q2 (contains 50% of the data).

The interval 3 - 10 covers two quarters (Min to Q2), meaning 50% of the data is contained in this range.
Answer: 3 - 10

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(g) What value could represent the 53rd percentile?

• The median (50th percentile) is 10.
• The 53rd percentile will be slightly above the median, closer to Q3.
  Thus, 11 is a reasonable estimate.

Answer: 11

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Let me know if you have any further questions or need clarification!

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Here are five related questions to expand your understanding:

1. How do you interpret outliers in a box plot?
2. What does it mean if the IQR is very small?
3. Can two different datasets have the same five-number summary?
4. What other measures of spread exist besides the IQR?
5. How does skewness affect the shape of a box plot?

Tip: The IQR is often used to detect outliers by checking if any data point lies more than 1.5 times the IQR away from Q1 or Q3.