Reading a Breakfast-and-Sports Survey
At Peachtree Ridge Middle School, a wellness committee surveys 250 eighth graders during one lunch period about two yes-or-no questions: whether they eat breakfast on a typical school day, and whether they play on a school sports team. The results: 84 students eat breakfast and play a sport; 56 eat breakfast and do not play a sport; 36 skip breakfast and play a sport; 74 skip breakfast and do not play a sport. The physical education teacher reads the table and announces that “eating breakfast makes students about 27 percentage points more likely to play a sport,” then proposes a district-wide breakfast program. The principal wants to know what the conditional relative frequencies actually show, whether the survey design limits the conclusions, and whether these two variables appear to be associated at all.
1. Compute the conditional relative frequency of playing a sport for breakfast eaters and for breakfast skippers. Is the teacher's 27-percentage-point figure supported?
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Build the two groups from the cells.
- Breakfast eaters: 84 + 56 = 140 students. Sport rate = 84 ÷ 140 = 0.60, or 60%.
- Breakfast skippers: 36 + 74 = 110 students. Sport rate = 36 ÷ 110 ≈ 0.327, or about 33%.
The difference is 60% − 33% ≈ 27 percentage points, so the teacher's arithmetic is correct. The overall sport rate (120 ÷ 250 = 48%) sits between the two group rates, as it should.
Because the two conditional relative frequencies differ substantially, the data do suggest an association between the two variables in this sample: knowing a student's breakfast habit changes the predicted probability of sport participation. What the numbers cannot do is establish which variable influences the other.
2. The teacher predicts a district-wide breakfast program will raise sports participation by 27 percentage points. Evaluate that prediction using the survey and its design.
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The table comes from one observational survey, and 27 points is a difference between groups, not a measured effect of an intervention. Several problems block the causal claim.
- Confounding: a third variable such as family schedule, household income, or morning bus timing could drive both breakfast eating and sport participation, producing the association without either causing the other.
- Reverse direction: students with morning practice may eat breakfast because of sports, not the other way around.
- Sampling: all 250 students were surveyed during a single lunch period at one school, so students who skip lunch, are absent, or attend other schools are underrepresented; the sample may not represent the district.
A defensible statement is: in this sample, sport participation was about 27 percentage points higher among breakfast eaters, which suggests an association worth studying. Claiming a program would cause a 27-point gain goes beyond the data.
3. Of the students who play a sport, what fraction eat breakfast? How does that conditional relative frequency differ from the one computed for breakfast eaters, and why does mixing them up distort the conclusion?
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Sport players total 84 + 36 = 120 students. Of those, 84 eat breakfast: 84 ÷ 120 = 0.70, or 70%.
This is a different conditional relative frequency from the 60% in the earlier task, because the denominator changed.
- P(sport | breakfast) = 84 ÷ 140 = 0.60 - the base group is breakfast eaters.
- P(breakfast | sport) = 84 ÷ 120 = 0.70 - the base group is sport players.
Both are correct but answer different questions. Reversing them would let someone claim “70% of athletes eat breakfast, so breakfast is what makes them athletes,” which overstates the case. The joint relative frequency, 84 ÷ 250 = 0.336, says that about 34% of surveyed students both eat breakfast and play a sport, and it is not a conditional probability at all. Stating which denominator is in use is the safeguard against misleading comparisons.