Bus Riders and Sports: Conditional Probability and Sampling
A school wellness committee surveys 140 students about bus riding and after-school sports. Of the 80 students who ride the bus, 30 participate in a sport. Of the 60 students who do not ride the bus, 20 participate in a sport. A parent notes that 30/80 = 37.5% of bus riders play a sport, while 30/50 = 60% of athletes ride the bus, and concludes that riding the bus makes students less likely to play sports. The committee also learns that the survey was conducted by asking students in the cafeteria during one lunch period. Members want to know whether the parent's comparison is valid, what the relevant conditional probabilities are, and how the survey method affects conclusions. They plan to present a short explanation to the PTO.
1. Compute P(sport | bus), P(sport | no bus), and P(bus | sport). Explain what each quantity means and whether the parent's conclusion follows from them.
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P(sport | bus) = 30/80 = 0.375, or 37.5%. Among bus riders, 37.5% participate in a sport.
P(sport | no bus) = 20/60 = 1/3 ≈ 0.333, or about 33.3%. Among students who do not ride the bus, about 33.3% participate in a sport.
P(bus | sport) = 30/50 = 0.60, or 60%. Among students who participate in a sport, 60% ride the bus. The denominator here is the 50 total athletes: 30 bus riders plus 20 non-riders.
The parent compared P(sport | bus) with P(bus | sport), which are different conditional probabilities with different denominators. That comparison cannot support the conclusion that bus riding makes students less likely to play sports. The fair comparison is P(sport | bus) versus P(sport | no bus): 37.5% versus about 33.3%. In this sample, bus riders were slightly more likely, not less likely, to report a sport. This is an association, not proof of causation.
2. Identify the sampling method used and at least two ways it could bias the results. Propose a better sampling plan.
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The committee used a convenience sample: students who happened to be in the cafeteria during one lunch period. That group may not represent the whole school.
Possible biases include excluding students who eat elsewhere, leave campus, have a different lunch period, are absent, or avoid the cafeteria. The cafeteria group may also overrepresent or underrepresent bus riders or athletes if those groups have different lunch schedules, practice schedules, or dismissal routines. Students who choose to respond may differ from those who do not.
A better plan is to obtain a student roster and use a simple random sample or a stratified random sample by grade and bus status. The committee should survey all selected students, provide an anonymous response method, follow up with nonrespondents, and report the response rate. Random selection gives every student a known chance of being included and reduces selection bias.
3. Suppose the committee wants to know if bus riders are less likely to play sports. Write a conclusion in context that uses the computed conditional probabilities and acknowledges the study design.
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In this sample, bus riders were somewhat more likely to report participating in a sport than non-riders: 30 of 80 bus riders, or 37.5%, compared with 20 of 60 non-riders, or about 33.3%. The difference is about 4.2 percentage points. Therefore the data do not support the claim that bus riders are less likely to play sports.
However, the survey used a convenience sample from one cafeteria lunch period, so the results may not generalize to the school. The observed difference could be due to sampling bias or other confounding factors such as grade level, schedule, or access to transportation. A random sample would be needed before drawing a schoolwide conclusion, and even a random sample would show association rather than causation.