Sleep and Breakfast Survey
A school health team surveys 200 students during first lunch about sleep and breakfast. Results: 70 students slept at least 7 hours and ate breakfast; 30 slept at least 7 hours and skipped breakfast; 40 slept fewer than 7 hours and ate breakfast; 60 slept fewer than 7 hours and skipped breakfast. The team concludes that sleeping at least 7 hours makes students more likely to eat breakfast, and proposes a school-wide sleep campaign. A teacher notes that the survey was given only to students who arrived by 7:45 a.m. and only during first lunch. The team asks you to analyze the association, evaluate the causal claim, and recommend a better sampling plan.
1. Build the two-way table and compute P(breakfast | slept at least 7 h) and P(breakfast | slept fewer than 7 h). Describe the association in context.
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Two-way table:
| Ate breakfast | Skipped breakfast | Total | |
|---|---|---|---|
| Slept ≥7 h | 70 | 30 | 100 |
| Slept <7 h | 40 | 60 | 100 |
| Total | 110 | 90 | 200 |
P(breakfast | ≥7 h) = 70/100 = 0.70. P(breakfast | <7 h) = 40/100 = 0.40.
The difference is 0.30, and the ratio is 0.70/0.40 = 1.75. In this sample, students who slept at least 7 hours were 1.75 times as likely to eat breakfast.
2. Does the survey support the claim that sleeping at least 7 hours makes students more likely to eat breakfast? Identify at least two specific weaknesses in the design.
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The data show an association, not proof of causation. This was an observational survey, so other variables could explain both sleep and breakfast: morning schedule, work or caregiving, sports, grade level, or general health habits. The sampling frame is also biased: only students who arrived by 7:45 a.m. and were in first lunch were included. Early arrivers may be more likely to have stable morning routines and to eat breakfast, which could inflate the 0.70 estimate or distort the comparison. Self-reported data may also be inaccurate. A causal claim would need a randomized or well-controlled study, or at least a random sample with confounders measured.
3. A second random sample of 150 students finds P(breakfast | ≥7 h) = 0.62 and P(breakfast | <7 h) = 0.48. Does the new evidence still support an association? Does it support a causal claim? Explain.
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New sample: P(breakfast | ≥7 h) = 0.62, P(breakfast | <7 h) = 0.48. Difference = 0.14; ratio = 0.62/0.48 ≈ 1.29. An association still appears in the new sample, but it is weaker than the original 0.30 difference. The new evidence does not establish causation for the same reason: it is still observational, and the sampling method matters. Before a school-wide campaign, the team should draw a random sample from the full roster, survey students at different times, and record possible confounders. The campaign may still be worthwhile, but the claim should be worded as “associated with” rather than “causes” unless a stronger design is used.