Interpreting a Student Well-Being Survey
A university surveys 1,200 randomly selected students about high stress. 300 students respond (25%). The respondent breakdown is: medicine 100, engineering 50, humanities 60, business 50, other 40. The university's student population is: medicine 10%, engineering 30%, humanities 25%, business 20%, other 15%. Among respondents, the share reporting high stress is: medicine 80%, engineering 40%, humanities 50%, business 55%, other 62.5%. The overall respondent figure is about 60.8%. A news office plans to report '60% of students report high stress.' A methods instructor asks whether that headline is defensible, given the response pattern. The survey had one reminder email and no incentives. Assume the reported stress rates within each faculty are accurate for respondents, but response propensity differs by faculty. The question is whether the raw 60.8% should be adjusted and what the adjusted estimate would be.
1. Identify the main source of bias and explain why the raw 60.8% may mislead.
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The main problem is nonresponse bias, not random sampling error. Response rates differ by faculty: medicine is 100/300 = 33.3% of respondents but only 10% of the population; engineering is 50/300 = 16.7% of respondents but 30% of the population. Medicine respondents report the highest stress rate (80%), while engineering reports the lowest (40%). Because the high-stress group is overrepresented and the low-stress group is underrepresented, the raw 60.8% is pulled upward. A 25% response rate alone does not prove bias, but the faculty-specific response pattern does.
2. Compute a faculty-weighted estimate using population shares. Show the arithmetic.
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Use the population shares as weights: Weighted estimate = 0.10×80 + 0.30×40 + 0.25×50 + 0.20×55 + 0.15×62.5 = 8 + 12 + 12.5 + 11 + 9.375 = 52.875%, or about 53%. This is lower than the raw 60.8% because it gives engineering its correct 30% weight and reduces medicine's weight from 33.3% to 10%.
3. What should the news office report, and what caveat should accompany it? Is weighting sufficient?
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The news office should report about 53%, not 60.8%, with a caveat: 'After adjusting for faculty composition, about 53% of respondents reported high stress; the survey had a 25% response rate, so nonresponse bias may remain.' Weighting is not a complete fix. It corrects only the observed faculty mix. If students who respond within each faculty differ in stress from nonrespondents-for example, more stressed students may be more likely to answer-the weighted estimate is still biased. The office should avoid causal or universal claims and should report the response rate and weighting method. If the university needs a defensible prevalence estimate, it should use incentives, more reminders, or a short nonresponse follow-up.