Confounding by Track in a Sport and Grades Survey
A school newsletter reports that 60% of students who play a sport have a grade average above 80, compared with 45% of students who do not. The sample contains 200 athletes recruited from team lists and 150 non-athletes recruited from a study hall. Among athletes, 80% are in the advanced track; among non-athletes, 50% are. In the advanced track, 70% of athletes and 70% of non-athletes exceed 80. In the regular track, 20% of athletes and 20% of non-athletes exceed 80. A parent concludes that playing a sport raises a student's chance of a high average by 15 percentage points and urges all students to join a team.
1. Verify the newsletter's 60% and 45% from the subgroup rates. What does the comparison of identical within-track rates imply about the parent's causal claim?
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Athletes: 0.80 × 0.70 + 0.20 × 0.20 = 0.56 + 0.04 = 0.60, or 60%. Non-athletes: 0.50 × 0.70 + 0.50 × 0.20 = 0.35 + 0.10 = 0.45, or 45%. The reported percentages follow from the subgroup rates.
However, within each track the rates are identical for athletes and non-athletes: 70% in advanced and 20% in regular. The 15-point overall gap is therefore a composition effect: athletes are much more likely to be in the advanced track (80% versus 50%). Track is a confounder, so the parent's causal claim is not supported. The convenience sample from team lists and a study hall also limits generalization to all students.
2. The school wants a defensible estimate of the proportion of all students with average above 80, with a 95% margin of error no greater than 3 percentage points. Using conservative p = 0.5, calculate the required sample size and explain one change to the sampling method.
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Using n = (z/E)² p(1 − p) with z = 1.96, E = 0.03, and p = 0.5: n = (1.96 ÷ 0.03)² × 0.25. Now 1.96 ÷ 0.03 = 65.333; squaring gives about 4268.4; multiplying by 0.25 gives about 1067.1. Round up to at least 1068 students.
One improvement is to take a random sample of all students, or a stratified random sample with separate random samples from the advanced and regular tracks and weighted estimates. This avoids the team-list/study-hall convenience sample and allows track-specific reporting rather than a misleading overall comparison.