Bike-to-School Survey and a Principal's Claim
A student council surveys 200 classmates about biking to school. In grade 7, 45 of 100 students say they bike; in grade 8, 25 of 100 say they bike. The principal reads a draft report stating, 'Only 25% of students bike to school, and eighth graders are much less likely to bike than seventh graders.' The principal wants to use the report to decide whether to request a bike rack expansion. A council member points out that the 25% figure came from the eighth-grade group, not all students. The school has 200 students total, and the grant program requires at least 40% of students to bike. The council also plans a follow-up survey of 20 students and is considering asking only students at the bike rack before school. Before submitting the report, the council must correct the overall rate, evaluate the association between grade level and biking, and judge the follow-up survey plan.
1. Correct the overall biking rate and evaluate the principal's statement about eighth graders.
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The overall bike rate is the total number of bikers divided by the total number surveyed: (45 + 25)/200 = 70/200 = 0.35, or 35%.
The principal's 25% is the eighth-grade rate only: 25/100 = 25%. The statement that eighth graders are much less likely to bike than seventh graders is supported by the sample: 45% of seventh graders bike versus 25% of eighth graders, a difference of 20 percentage points.
The report should say '35% of students overall bike' and separately report the grade-level rates, rather than labeling 25% as the schoolwide rate.
2. Use conditional relative frequencies to decide whether grade level and biking are associated. What does the grant threshold require?
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Conditional relative frequencies: P(bike | grade 7) = 45/100 = 45%; P(bike | grade 8) = 25/100 = 25%. Because these rates differ substantially, grade level and biking status are associated in this sample. That does not prove grade level causes biking; distance from school, safety, and family rules could also explain the difference.
For the grant, 40% of 200 students is 80 students. The school currently has 70 bikers, so it needs 10 more biking students to reach the 40% threshold, assuming the total stays at 200.
3. Judge the plan to survey 20 students only at the bike rack. What sampling method would be better and why?
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Surveying only students at the bike rack is biased: it almost guarantees a high biking rate and misses students who walk, ride a bus, or are driven. Those students may be less likely to bike, so the follow-up estimate would probably be too high.
A better plan is to randomly select 20 students from the full list of 200, or to use a stratified sample of 10 randomly chosen seventh graders and 10 randomly chosen eighth graders. Random selection reduces bias and lets the council compare grade-level rates fairly. The council should also ask the same question in the same way to every selected student.