Sports and Math Pass Rates: Conditional Relative Frequency and Causation
A student council surveys 200 students at a high school about whether they play a school sport and whether they passed last year's math course. The results are: 72 students play a sport and passed math; 18 play a sport and did not pass; 84 do not play a sport and passed; 26 do not play a sport and did not pass. A council member notes that 80% of students who play a sport passed math, compared with about 76% of students who do not play a sport, and concludes that playing a sport causes students to pass math. The council wants to use this conclusion to support a proposal requiring all students to join a sport. Another member points out that the survey was given only to students in one lunch period and asks whether the data support the cause-and-effect claim and whether the sample represents the school.
1. Build a two-way table and calculate the conditional relative frequencies of passing math for students who play a sport and for students who do not. Round to the nearest tenth of a percent.
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The two-way table is:
| Passed math | Did not pass | Total | |
|---|---|---|---|
| Plays a sport | 72 | 18 | 90 |
| Does not play a sport | 84 | 26 | 110 |
| Total | 156 | 44 | 200 |
For students who play a sport, the conditional relative frequency of passing is
[\frac{72}{90}=0.8=80.0%.]
For students who do not play a sport, the conditional relative frequency of passing is
[\frac{84}{110}\approx 0.7636=76.4%.]
The pass rate is 3.6 percentage points higher for students who play a sport. The overall pass rate is (\frac{156}{200}=78%), which lies between the two conditional rates.
2. Does the data show an association between playing a sport and passing math? Explain using the conditional relative frequencies, and explain why the difference does not prove causation.
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There is a weak positive association: 80.0% of students who play a sport passed math, compared with 76.4% of students who do not play a sport. Since these conditional pass rates differ, the two variables are associated in this sample.
However, association does not establish causation. A third variable could affect both sports participation and math passing, such as attendance, prior achievement, motivation, study habits, or family support. The direction could also be reversed: students may need to pass math to remain eligible for sports, so passing math helps them play rather than playing helping them pass. The difference of 3.6 percentage points is also small and could be due to sampling variability. To support a cause-and-effect claim, the council would need a randomized experiment or a carefully controlled longitudinal study, not just an observational survey.
3. The survey was given only during one lunch period. Explain how this affects the council's claim that requiring sports will improve math pass rates, and describe one change to the survey design that would produce more representative data.
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A single lunch period may not represent the whole school. Students are assigned to lunch waves by schedule, grade level, or course load, so one lunch period could overrepresent or underrepresent certain groups. Students who eat elsewhere, leave campus, or are in a club or tutoring session during that lunch may be missed entirely. If the sample is biased, the 80.0% and 76.4% pass rates may not estimate the school's true rates, and the conclusion is even weaker.
A better design would take a random sample from the entire student roster and survey students across multiple lunch periods and days, checking that all grade levels are included and tracking nonresponse. Even with a representative sample, the data would show association, not causation; to test whether requiring sports improves math pass rates, the school would need a randomized or controlled study.