Breakfast and Tardiness Survey
A student wellness committee surveyed 240 students about breakfast habits and tardiness. Of the 90 students who reported eating breakfast at least four times a week, 18 had been late to first period at least twice in the past month. Of the 150 students who reported eating breakfast less often, 45 had been late at least twice. The committee plans to recommend a before-school breakfast program, claiming that 'students who eat breakfast are half as likely to be late.' A reporter asks whether the survey proves that eating breakfast causes lower tardiness. The survey had a 68% response rate, and students who responded may differ from those who did not. No students were randomly assigned to breakfast conditions.
1. Compute the conditional probabilities of being late for the two groups and the overall late rate.
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Late rate among breakfast eaters = 18/90 = 0.20, or 20%. Late rate among others = 45/150 = 0.30, or 30%. Overall = (18 + 45)/(90 + 150) = 63/240 = 0.2625, or 26.25%. The breakfast group has a lower observed late rate by 10 percentage points.
2. Evaluate the claim that breakfast eaters are 'half as likely' to be late.
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The ratio of late rates is 0.20/0.30 = 2/3 ≈ 0.67, so breakfast eaters are about two-thirds as likely to be late, a 33% relative reduction, not 50%. The claim 'half as likely' would require a rate near 15% in the breakfast group, not 20%. The absolute difference is 10 percentage points, which is meaningful but smaller than the claim suggests.
3. Explain whether the survey supports a causal conclusion and describe one way response bias could affect the result.
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No. This is an observational survey, not a randomized experiment. Students who eat breakfast may differ in other ways, such as sleep habits, morning routines, or transportation, that also affect tardiness, so association does not prove causation. Response bias is possible: only 68% responded, and if frequently tardy students were less likely to return the survey, the measured late rates could be too low, or the gap between groups could change. A causal claim would need a controlled study.