A Study-Hours Survey That Overreaches
A student researcher wants to estimate how many hours Tubman University students study each week. On a Tuesday from 8:00 p.m. to 10:00 p.m., she stands at the main library entrance and surveys 60 students who enter. The mean is 22 hours per week. She writes in her report: 'The average Tubman student studies 22 hours weekly.' Enrollment is 4,200, of whom 1,100 live on campus. A separate random sample of 120 on-campus residents gives a mean of 14 hours and a standard deviation of 6 hours. In the library sample, 18 of 60 respondents were first-year students, although first-years make up 35 percent of enrollment. Her instructor asks her to check the claim before the report goes to the student senate.
1. Identify the main bias in the library survey and explain its direction and consequences for the claim.
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Main bias: selection (coverage) bias. The sampling frame is 'students entering the main library between 8 and 10 p.m. on a Tuesday.' That is not the student body. Students who study in dormitories, at home, in laboratories, or not at all are excluded, and those present at the library late on a weeknight are plausibly more studious than average.
Direction: upward. The reported 22 hours likely overestimates the campus mean. The 30 percent first-year share versus 35 percent in enrollment is a smaller problem, but it confirms the sample is not representative.
Other limitations: 60 respondents is a small sample with a wide margin of error; self-reported study time is subject to recall and social-desirability bias; students who decline to answer are not recorded.
Consequence: the sentence 'the average Tubman student studies 22 hours weekly' is unsupported. It should read: 'among 60 students entering the library on one Tuesday evening, the mean reported study time was 22 hours' - a description of the sample, not an estimate for the university.
2. Using the dormitory sample, compute a 95 percent confidence interval and state precisely what it means.
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Calculation. Standard error = s ÷ √n = 6 ÷ √120. Since √120 ≈ 10.95, SE ≈ 0.548 hours. The 95 percent margin is 1.96 × 0.548 ≈ 1.07 hours. Interval: 14 ± 1.07, or about 12.9 to 15.1 hours.
Interpretation. If the same sampling procedure were repeated many times, about 95 percent of intervals constructed this way would contain the true mean study time of on-campus residents. It is not a statement that 95 percent of students study between 12.9 and 15.1 hours.
Caveats. The interval assumes a simple random sample and approximate normality, which is reasonable at n = 120. It covers only on-campus residents, who are 1,100 of 4,200 students. It says nothing about off-campus students, and because it relies on self-report, the true mean could still be biased even though the interval is statistically precise.
Useful contrast: the library sample's 22 hours lies far outside this interval, which is evidence that the library sample is not measuring the same population.
3. The researcher proposes weighting the two results: 14 hours for the 1,100 on-campus students and 22 hours for the 3,100 off-campus students. Compute the estimate and explain why it is still not trustworthy.
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Weighted estimate. (1,100 × 14 + 3,100 × 22) ÷ 4,200 = (15,400 + 68,200) ÷ 4,200 = 83,600 ÷ 4,200 ≈ 19.9 hours.
Why it is still not trustworthy.
- Weighting corrects composition, not measurement. It assumes the 22-hour figure validly estimates off-campus students, but that figure comes from the biased library sample. A weighted average of one good estimate and one biased estimate remains biased.
- The stratum means come from different designs. The dorm figure is a random sample; the off-campus figure is a convenience sample from one place and time.
- It assumes all 1,100 on-campus students are represented by the dorm sample and that all 3,100 off-campus students are alike, ignoring differences by year, programme, and employment.
Better design: draw a simple random or stratified sample from the registrar's enrolment list, stratify by residence and year, contact students at several times of day, and use a short diary or app log rather than a single recall question. Report the design and response rate alongside the estimate.