Two samples of students were asked to indicate the number of email messages they sent in the last 24 hours. The responses given by each sample are shown in the table below.

Which one of the following statements about these samples is correct?

(Note that if you have difficulties seeing the above table, you can download it here
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Question

Two samples of students were asked to indicate the number of email messages they sent in the last 24 hours.  The responses given by each sample are shown in the table below.

Which one of the following statements about these samples is correct?

(Note that if you have difficulties seeing the above table, you can download it here 
Download here
)

BlackTom AI · Accepted Answer

To assess the statements about the two samples, I will first compute the key descriptive statistics for each sample.
Sample #1 data (11 values): 0, 18, 7, 12, 9, 16, 1, 22, 34, 10, 5. Sort them: 0, 1, 5, 7, 9, 10, 12, 16, 18, 22, 34. The median is the 6th value, which is 10. For the IQR, the lower half (first 5) is 0, 1, 5, 7, 9 with median 5 (Q1), and the upper half (last 5) is 12, 16, 18, 22, 34 with median 18 (Q3). IQR = 18 − 5 = 13.
Sample #2 data (11 values): 13, 8, 1, 4, 25, 12, 14, 18, 3, 0, 23. Sort them: 0, 1, 3, 4, 8, 12, 13, 14, 18, 23, 25. The median is the 6th value, which is 12. For the IQR, the lower half (first 5) is 0, 1, 3, 4, 8 with median 3 (Q1), and the upper half (last 5) is 13, 14, 18, 23, 25 with median 18 (Q3). IQR = 18 − 3 = 15.
With these values, we can evaluate each statement:

Option 1: "Sample #1 has a smaller IQR and a smaller median than Sample #2."
- Compare medians: 10 (Sample #1) vs 12 (Sample #2). Sample #1 has a smaller median, which matches the claim.
- Compare IQRs: 13 (Sample #1) vs 15 (Sample #2). Sample #1 has a smaller IQR, which also matches the claim.
- Therefore, this option correctly describes both statistics for the two samples.

Option 2: "Sample #1 has a smaller IQR and a larger median than Sample #2."
- This would require Sample #1 to have a larger median, but we found Sample #1 has median 10 versus 12 for Sample #2, so the median is not larger. This option is inconsistent with the data.

Option 3: "Sample #1 has a larger IQR and a larger median than Sample #2."
- Both the median and IQR would need to be larger for Sample #1, but the median is smaller (10 vs 12) and the IQR is smaller (13 vs 15). This option is false.

Option 4: "Sample #1 and Sample #2 have the same IQR and the same median."
- The medians are 10 and 12, not equal, and IQRs are 13 and 15, not equal. This option is incorrect.

Option 5: "Sample #1 has a larger IQR and a smaller median than Sample #2."
- This would require a larger IQR for Sample #1 (not the case) while the median is smaller (which is true). Since the IQR portion is false, this option is not supported by the data.

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