Start-Time Survey: Stratification, Weighting, and Margin of Error
The student council at a high school with 900 students wants to know whether students support moving the start time 30 minutes later. They survey 80 students: 48 support and 32 oppose. However, 30 of the surveys came from the honors lunch period, which has 180 students total, and 50 came from the general lunch period, which has 720 students. Among honors respondents, 21 support; among general respondents, 27 support. The council plans to announce that 60% of students support the change, with a margin of error of about 11 percentage points. A teacher points out that the honors lunch was surveyed on a day when a field trip reduced attendance, and that 16 of the 96 students originally selected did not return surveys.
1. Compute the overall, honors, and general support rates. Explain why the raw 60% is misleading.
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Overall support = 48/80 = 0.60 = 60%. Honors support = 21/30 = 0.70 = 70%. General support = 27/50 = 0.54 = 54%. The honors group is overrepresented: it is 30/80 = 37.5% of the sample but only 180/900 = 20% of the school. Honors respondents support at a higher rate, so the raw 60% is pulled upward. This is a stratification or sampling-bias problem, not evidence that the school as a whole supports at 60%. The field-trip absence and the 16 nonresponses could add further bias if those missing students differ from respondents.
2. Use the actual population shares to produce a weighted estimate. Evaluate the claim that 60% of students support the change, using the stated margin of error.
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Weight by actual population shares: honors 180/900 = 0.20, general 720/900 = 0.80. Weighted estimate = 0.20(0.70) + 0.80(0.54) = 0.14 + 0.432 = 0.572, or 57.2%. The stated margin of error is about 11 points, giving roughly 46.2% to 68.2%. The claim of 60% is inside that interval, but so is a value below 50%; with this sample design the council cannot confidently claim majority support. The nonresponse adds further uncertainty if nonrespondents differ from respondents, so the announcement should not state 60% as a firm fact.
3. Recommend a better survey design and sample size if the council wants a margin of error near 5 percentage points.
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A better design is a stratified random sample: randomly select about 20% from honors and 80% from general, or take a simple random sample of all 900 with follow-up to reduce nonresponse. Report the weighting method and the response rate. The rough margin-of-error formula 1/sqrt(n) gives about 11% at n=80; to get about 5% you need n near 400, since 1/sqrt(400) = 0.05. The council should also avoid surveying only one lunch period on an unusual field-trip day, use a neutral question, and follow up with nonrespondents so the sample better represents the school.