Two-Way Table on Breakfast and Tardiness with a Biased Sample
The wellness committee surveyed 240 students about breakfast and tardiness. Students were surveyed in the cafeteria breakfast line between 7:40 and 7:55 a.m. The results are in the two-way table below. The principal wants to use the data to argue for a later school start, claiming that 'skipping breakfast makes students 20% more likely to be late.' A committee member notes that students who skip breakfast are unlikely to be standing in the breakfast line, so the sample may not represent the school. The school has 1,100 students. The committee must write a short summary that reports the relevant conditional relative frequencies, evaluates the principal's 20% claim, and recommends whether the survey should be repeated before any policy decision is made.
| Late this month | Not late | Total | |
|---|---|---|---|
| Eats breakfast regularly | 18 | 102 | 120 |
| Skips breakfast | 42 | 78 | 120 |
| Total | 60 | 180 | 240 |
1. Compute the conditional relative frequency of being late for students who eat breakfast and for students who skip breakfast. Also compute the overall late rate. What do these numbers suggest about association in this sample?
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For breakfast eaters: 18 ÷ 120 = 0.15, or 15% were late. For breakfast skippers: 42 ÷ 120 = 0.35, or 35% were late. Overall: 60 ÷ 240 = 0.25, or 25% were late.
The late rate is 20 percentage points higher among skippers (35% − 15% = 20%). As a ratio, 35 ÷ 15 ≈ 2.33, so skippers in this sample were about 2.33 times as likely to be late. This is evidence of an association between the two variables in the sample. It does not by itself show that breakfast causes punctuality, and it does not address whether the sample represents the school.
2. Evaluate the principal's claim that skipping breakfast makes students '20% more likely' to be late. Identify both the numerical problem and the causation problem.
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Numerically, 35% − 15% = 20 percentage points, not 20%. A 20% relative increase from 15% would be 15% × 1.20 = 18%, which is far below the observed 35%. The correct relative increase is (35 − 15) ÷ 15 = 20 ÷ 15 ≈ 1.33, or about 133% more likely, which is the same as saying about 2.33 times as likely.
Causally, the word 'makes' claims causation, but this is an observational survey. Other variables could explain both behaviors: unreliable transportation, earlier work shifts, caregiving duties, or inconsistent supervision. Without a controlled or well-designed observational study that accounts for such factors, the data support association at most, not cause and effect.
3. The survey was conducted in the breakfast line. Explain how this sampling method could distort the association, and recommend a better design. Address whether the 240 responses can be treated as a random sample of the 1,100 students.
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The sample is not random. Students in the breakfast line are, by location, more likely to be breakfast eaters, so the 120-to-120 split almost certainly overrepresents breakfast eaters and underrepresents skippers. The true proportion of skippers in the school is probably higher, and the students who skip breakfast but still arrive early enough to be in the cafeteria may differ from skippers generally, which can distort the within-group late rates in either direction.
Because selection depended on being in a particular place at a particular time, the 240 responses cannot be treated as a random sample of all 1,100 students, and no margin of error or school-wide percentage should be reported from them.
A better design: draw a random sample from the enrollment list, survey across all grades and both arrival times, use an anonymous questionnaire not tied to the cafeteria, and define 'regularly' and 'late' in advance. If the association persists in a random sample, it is still not proof of causation, but it would be far more defensible.