Later Start Survey: Conditional Percents and a Convenience Sample
The student council surveyed 200 students about moving the start of the school day 30 minutes later. Because the council could reach students only during first period, the survey was given to six first-period classes chosen for convenience: 120 underclassmen (grades 9–10) and 80 upperclassmen (grades 11–12). Results: 84 of the 120 underclassmen support the change, and 28 of the 80 upperclassmen support it. The school newspaper runs the headline '56% of students back later start,' reporting that 112 of 200 respondents said yes. A council member notes that the sample is large, so 'the margin of error is only about 7 points and the majority is statistically solid.' A separate poll of 10 students about nightly homework found a median of 1.5 hours and a mean of 2.0 hours.
1. Compute the support rate for each grade group and for the full sample. Does grade level appear to be associated with support? Explain what comparison justifies your answer.
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Conditional (row) percentages:
- Underclassmen: 84/120 = 70%
- Upperclassmen: 28/80 = 35%
- Overall: 112/200 = 56%
The gap is 35 percentage points, which is large relative to the group sizes - the two groups behave very differently, so grade level does appear to be associated with support.
The aggregate is a weighted average of the two conditional rates: (120/200)(70%) + (80/200)(35%) = 42% + 14% = 56%.
That decomposition is the key insight: because underclassmen make up 60% of the sample, the 56% headline mostly reflects their views. The aggregate does not describe any actual group - it is a blend of a strong-majority group and a strong-minority group. Any recommendation should report the two rates separately; a single 56% figure could mislead the administration into thinking support is uniform.
2. Evaluate the newspaper headline and the 'margin of error is only about 7 points' claim. Identify the specific flaw in the sampling design and explain what a margin of error would and would not tell you here.
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The headline overreaches. 56% is a sample statistic from 200 respondents, not the share of the student body.
The design flaw is a convenience sample, not a random one. Surveying six first-period classes excludes students who arrive late, are absent, or have first period off campus - plausibly exactly the students most interested in a later start. Students in the same class also share a teacher and a time slot, so their responses are not independent (a cluster effect), which inflates the effective margin of error beyond the textbook formula.
The stated margin of error is misapplied. The ±7 points comes from roughly 1/√200 ≈ 3.5% doubled for 95% confidence, and it assumes a simple random sample. Here it would understate uncertainty because it captures only sampling variability, not bias. A margin of error cannot repair a sample that systematically misses part of the population - a large biased sample can be confidently wrong.
A defensible design: a stratified random sample by grade with a stated response rate and follow-up for non-responders.
3. The homework poll found a median of 1.5 hours and a mean of 2.0 hours. Explain how both can be true, describe the likely shape of the distribution, and state which statistic the council should quote as typical.
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The two statistics answer different questions. The mean uses every value, so extreme values pull it; the median is the middle value and is resistant to extremes.
A concrete data set with these summaries: 0.5, 1, 1, 1.5, 1.5, 1.5, 2, 2, 3, 6.
- Sorted, the middle two values are 1.5 and 1.5, so the median = 1.5 h.
- Sum = 20, so the mean = 2.0 h.
The single 6-hour report sits 4.5 hours above the median and alone pulls the mean up by 0.4 h. The distribution is right-skewed: most students cluster near 1–2 hours with a long tail of heavy homework nights.
Quote the median (1.5 h) as the typical load, and report the mean alongside with a note about the tail so the average is not mistaken for a typical experience. Also flag that n = 10 is far too small to support any margin of error or population claim - one additional respondent could move the median.