Coffee, Sleep, and Study-Hour Summary
A student health survey at a small college asked 120 undergraduates two questions: whether they drink coffee daily and whether they usually get at least 7 hours of sleep. Among the 48 daily coffee drinkers, 18 reported at least 7 hours of sleep. Among the 72 students who do not drink coffee daily, 54 reported at least 7 hours of sleep. The survey also recorded self-reported nightly study hours for a random sample of nine respondents: 2, 3, 3, 4, 4, 4, 5, 6, and 12. A classmate claims that coffee drinking is unrelated to sleep duration and that the mean is the best summary of study time. The student wants to test both claims using conditional probabilities and measures of center, and to explain what the outlier does to the mean.
1. Calculate P(at least 7 h sleep | coffee) and P(at least 7 h sleep | no coffee). What do these suggest about the classmate’s independence claim?
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For coffee drinkers: 18 of 48 sleep at least 7 hours, so P(sleep ≥ 7 | coffee) = 18/48 = 0.375. For non-coffee drinkers: 54 of 72 sleep at least 7 hours, so P(sleep ≥ 7 | no coffee) = 54/72 = 0.75.
The overall rate is 72/120 = 0.60. If coffee drinking and sleep were independent, both conditional probabilities would equal the overall rate, or at least equal each other. They do not: the no-coffee group’s rate is twice the coffee group’s rate in this sample. That is evidence of an association, though the survey cannot establish causation. The classmate’s “unrelated” claim is not supported by these conditional probabilities.
2. If a respondent reports at least 7 hours of sleep, what is the probability they drink coffee daily? Compare with the unconditional probability of being a coffee drinker and explain why the two differ.
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The number sleeping at least 7 hours is 18 + 54 = 72. Of those, 18 drink coffee, so P(coffee | sleep ≥ 7) = 18/72 = 0.25.
The unconditional probability of being a coffee drinker is 48/120 = 0.40. These differ because the condition changes the denominator and restricts attention to good sleepers. In this sample, coffee drinkers are underrepresented among good sleepers: they make up 40% of all students but only 25% of those sleeping at least 7 hours. This is the same association seen from the other direction in the previous conditional probability; it does not mean coffee causes poor sleep.
3. Compute the mean and median of the nine study-hour values. Which better summarizes typical study time, and by how much does the outlier shift the mean?
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The nine values sum to 2 + 3 + 3 + 4 + 4 + 4 + 5 + 6 + 12 = 43. Mean = 43/9 = 4.777... ≈ 4.78 hours. With nine sorted values, the median is the fifth value: 4 hours.
The value 12 is an outlier. Removing it gives eight values summing to 31, with mean 31/8 = 3.875 hours. The outlier raises the mean by 4.777 - 3.875 ≈ 0.90 hours, while the median remains 4 hours with or without it. For a typical study time, the median is more resistant and more representative here; the mean is pulled upward by one unusually high response and overstates the center of the group.