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Let's analyze the problem:

**Given:**
- Susan's score is in the **82nd percentile**.
- Keith's score is in the **21st percentile**.
- There were over **13,000 students** who took the exam.

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### (a) Which of the following must be true about Susan's and Keith's scores?

- **Susan scored higher than Keith**: Since Susan's score is in the 82nd percentile and Keith's is in the 21st percentile, **Susan scored higher than Keith**.
  
- **Both Susan and Keith scored higher than the median**: The median represents the 50th percentile. Since Keith is in the 21st percentile, which is below the median, this is **false**.

- **Susan got 61 more questions correct than Keith**: The percentile gives a rank, not a direct comparison of how many more questions were answered correctly. This cannot be determined from the given information, so it's **false**.

- **Susan's and Keith's scores were both in the bottom half of all scores**: The bottom half would refer to below the 50th percentile. Since Susan is in the 82nd percentile, this statement is **false**.

The correct answer is: **Susan scored higher than Keith**.

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### (b) Which of the following must be true about Susan's score?

- **Susan scored higher than about 18% of the students**: This is **false** because Susan is in the 82nd percentile, which means she scored higher than 82% of the students.

- **About 18% of the students who took the exam scored higher than Susan**: This is **true**. If Susan is in the 82nd percentile, that means 18% of students scored higher than her.

- **Susan got about 82% of the questions correct**: The percentile rank does not directly correspond to the percentage of questions answered correctly, so this is **false**.

- **Susan got about 82% of the questions wrong**: Since Susan scored in the 82nd percentile (indicating a high score), this is **false**.

The correct answer is: **About 18% of the students who took the exam scored higher than Susan**.

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Let me know if you'd like more details!

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**Follow-up questions:**
1. How does percentile ranking differ from percentage scores in an exam?
2. What is the significance of the median in percentile distributions?
3. Could Susan's 82nd percentile be translated into a specific number of correct answers if the exam data were available?
4. What does being in the 21st percentile imply for Keith’s performance compared to the majority of students?
5. How would the interpretation change if Susan and Keith's scores were given in percentages instead of percentiles?

**Tip:** Remember, percentile ranks are relative positions compared to other students, not direct measures of the number of questions answered correctly!

Last year over 13,000 students took an entrance exam at a certain state university. Susan's score was at the 82nd percentile. Keith's score was at the 21st percentile. (a) Which of the following must be true about Susan's and Keith's scores? (b) Which of the following must be true about Susan's score?

Learn how to interpret percentile ranks in exam scores using a scenario where Susan's score is in the 82nd percentile and Keith's score is in the 21st percentile. This problem explores the meaning of percentile ranks, relative performance, and median comparisons.

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