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Question
a sociologist studying marriage customs collected data on the length of time, in months, between the proposal and wedding. the following boxplot and table give the five - number summary for couples in which at least one person was getting married for the first time and for couples in which both people had been previously married.
| period | minimum | q1 | median | q3 | maximum |
|---|---|---|---|---|---|
| previously married | 0.5 | 8 | 12 | 16 | 20 |
boxplot: first time married and previously married, x - axis: time (months) between proposal and wedding, range 0 - 40
based on the summary statistics, which of the following statements is supported by the boxplots and table?
a) the mean length of time for first - time couples is 18 months, and the mean length of time for previously married couples is 12 months.
b) there is a gap between 20 and 36 for the first - time married distribution.
c) because both distributions have a minimum of 0.5 months, the distributions are skewed to the left.
- Option A: The median (not mean) for first - time is 18 and previously married is 12. We can't conclude means from medians. Eliminate A.
- Option B: The first - time married distribution has a maximum of 36 and Q3 of 20. The boxplot shows a whisker from 20 to 36, but no gap (data exists there as it's a whisker). Wait, actually, the boxplot for first - time married has a whisker extending to 36, but between 20 (Q3) and 36, is there a gap? No, the whisker represents the range of non - outlier data. Wait, maybe misinterpretation. Wait, the table for first - time: Q3 = 20, Max = 36. The boxplot has a whisker from the box (up to Q3 or median? Wait, boxplot: box is from Q1 to Q3, median inside. Whiskers to min and max (non - outlier). So for first - time, Q3 is 20, max is 36. So the whisker goes from Q3 (20) to max (36), so there is no gap (data is present as the whisker is continuous). Wait, no, maybe the option is wrong? Wait, no, let's check other options.
- Option C: A left - skewed distribution has mean < median, and the tail is on the left. Here, both have min 0.5, but first - time: median 18, Q1 = 10, so the left tail (from min 0.5 to Q1 10) is shorter than the right tail (Q3 20 to max 36). So first - time is right - skewed. Previously married: median 12, Q1 = 8, Q3 = 16, max 20. Left tail (0.5 - 8) and right tail (16 - 20), left tail longer? Wait, no, 8 - 0.5 = 7.5, 20 - 16 = 4. So left tail longer, but min being same doesn't imply left skew. And first - time is right - skewed. So C is wrong.
Wait, maybe I made a mistake with B. Wait, the first - time married boxplot: the upper whisker goes from the box (Q3 = 20) to max = 36. But is there a gap between 20 and 36? No, the whisker is a continuous line, meaning data points are present (non - outlier). Wait, maybe the option is correct? Wait, no, the boxplot's whisker represents the range of data within 1.5IQR from Q1 and Q3. Wait, IQR for first - time: Q3 - Q1 = 20 - 10 = 10. 1.5IQR = 15. Q3+1.5IQR = 20 + 15 = 35. So max is 36, which is an outlier? Wait, the table says max is 36. So if 36 is an outlier, then between Q3 (20) and 35 (1.5IQR above Q3), there is a gap, and 36 is an outlier. Wait, maybe the option B is saying there is a gap between 20 and 36. If 36 is an outlier, then between 20 and 36 (excluding 36), there is a gap (no non - outlier data). So B is correct? Wait, let's re - evaluate:
- IQR for first - time: 20 - 10 = 10. Upper fence: Q3+1.5*IQR = 20 + 15 = 35. So data above 35 is an outlier. The max is 36, which is an outlier. So between Q3 (20) and 35, there is no data (since upper fence is 35), and then 36 is an outlier. So there is a gap between 20 and 36 (since 20 to 35 has no data, and 36 is an outlier). So B is correct.
Wait, but initially, I thought the whisker is to max, but if max is an outlier, the whisker should go to the largest non - outlier. So maybe the boxplot in the problem has a whisker to 35 (non - outlier) and then 36 is a dot (outlier). So between 20 (Q3) and 35 (end of whisker), there is data? No, whisker is from Q3 to the largest non - outlier. So if max is 36 (outlier), whisker is to 35, and 36 is an outlier. So between 20 and 35, there is data (the whisker), and 36 is separate. Wait, I'm confused. Let's check the other options again.
Option A: Median is 18 and 12, not mean. So A is wrong.
Option C: Skewness is not determined by minimum alone. First - time: median 18, Q1 = 10, Q3 = 20. The distance from median to Q1: 18 - 10 = 8; to Q3: 20 - 18 = 2. So left tail (min to Q1) is 10 - 0.5 = 9.5, right tail (Q3 to max) is 36 - 20 = 16. So righ…
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B. There is a gap between 20 and 36 for the first - time married distribution.