Survey Margin of Error and Generalization
A student council surveys cafeteria users about adding a salad bar. They randomly select 400 students from the 1,100 students who eat in the cafeteria on a typical day and invite them to an online poll; 320 respond. Of those, 198 say they would buy a salad at least once a week. The council reports 62% support with a margin of error of ±5 percentage points at 95% confidence. A member wants to claim that “a majority of all students at the school support the salad bar.” The school has 1,850 students total, including those who leave campus for lunch and therefore were not in the sampling frame.
1. Verify the reported 62% support using the response count, and give the 95% confidence interval implied by the margin of error.
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The response proportion is
$$\frac{198}{320}=0.61875\approx 61.9%,$$
which rounds to 62%.
A margin of error of ±5 percentage points gives the interval
$$62%-5%=57% \quad \text{to} \quad 62%+5%=67%.$$
So the plausible support level among the population actually sampled is roughly 57% to 67%.
2. Does the interval support the claim that a majority of the sampled population supports the salad bar? Explain.
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Yes, for the cafeteria-user population that the sample is meant to represent. The entire interval from 57% to 67% lies above 50%, so the data are consistent with majority support.
However, this conclusion depends on the margin-of-error model being appropriate. The 62% is based on the 320 respondents, not all 400 invited students, so nonresponse could bias the estimate if responders differ from nonresponders.
3. Can the council generalize the 62% result to all 1,850 students? Identify the main sampling-frame problem and one additional source of bias.
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Not directly. The sampling frame is the 1,100 cafeteria users, so students who leave campus for lunch had no chance of being selected. Generalizing to all 1,850 students assumes those excluded students have the same salad-bar preferences as cafeteria users, which is not justified.
A second issue is nonresponse: only 320 of 400 invited students responded. If students who like salad bars were more likely to respond, the 62% overstates support. A better design would randomly sample from all 1,850 students and follow up with nonrespondents.