Bus Survey Across Two Wards
A council survey asked residents whether they use the bus at least once a week. In North ward, 240 residents responded, and 45% said yes. In South ward, 160 responded, and 60% said yes. Response rates were 30% of 800 households in North and 40% of 400 households in South. A councillor says: 'South's higher response rate means the overall figure should be pulled toward South, and the combined weekly bus-use rate is 52.5%.' A resident counters that the survey is meaningless because the two wards have different response rates.
1. Evaluate the councillor's figure of 52.5%. Show the correct respondent-weighted calculation.
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52.5% is the unweighted average of 45% and 60%: (45 + 60) ÷ 2 = 52.5. That would be correct only if the two wards had equal numbers of respondents. They do not: North has 240 and South has 160. The counts are North 0.45 × 240 = 108 yes-responses and South 0.60 × 160 = 96. Combined = 204 of 400 = 51.0%, not 52.5%. The error is a base-size error: averaging percentages gives each ward equal weight regardless of how many people it represents. The 1.5-point difference is small but systematic, and the same mistake would be larger if the sample sizes were more unequal.
2. Assess the councillor's argument that South's higher response rate means the combined figure should be pulled toward South. Then give a defensible overall estimate, stating any assumption.
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Response rate tells us how much of the invited sample replied; it does not tell us how much to weight a ward in a combined individual-level estimate. The councillor is conflating representativeness with influence. If the aim is to estimate bus use among all households in the two wards, and if each ward's respondents are representative, the correct weights are the household populations: North 800 and South 400. The estimate is (0.45 × 800 + 0.60 × 400) ÷ 1200 = (360 + 240) ÷ 1200 = 50.0%. If the aim is only to describe the 400 respondents, 51.0% is correct. Neither figure justifies pulling toward South because its response rate is higher; a higher response rate may actually make South's 60% more trustworthy, but it does not change the arithmetic weight. The choice of 50% or 51% should be stated with its assumption.
3. Is the resident right that different response rates make the survey meaningless? Support your answer with a rough statistical check and identify what else would be needed.
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The resident is too strong. Different response rates reduce confidence and raise the possibility of non-response bias, but they do not make the survey meaningless. A rough check: the standard error of the difference is √(0.45×0.55/240 + 0.60×0.40/160) ≈ √(0.00103 + 0.00150) ≈ 0.050, so the 15-point gap is about three standard errors. If sampling were random, a gap that large would be unlikely to arise by chance. The real threat is that people who use the bus may be more or less likely to respond in each ward. To test that, we would need response-mode data, a short non-response follow-up, or demographic comparison with ward records. The resident's concern is a reason to check, not a reason to discard.