A Start-Time Survey With a Self-Selected Sample
A student council at a high school of 1,400 students wants to know whether students support moving the start time from 7:45 to 8:30. The council collects responses two ways: a table in the cafeteria during the first of three lunch waves, and a link posted on the school's Instagram account. They receive 186 responses, and 121 of them, or 65%, say yes. The same respondents are asked about sleep: the mean is 6.9 hours per night and the median is 7.2 hours. Fourteen students, mostly athletes with 5:45 a.m. practice, reported 4 hours or less. Support by grade comes out as 9th grade 58 of 88 in favor, 10th grade 41 of 62, 11th grade 17 of 26, and 12th grade 5 of 10. A council member drafts an announcement reading: "Support is strong at 65% and drops steadily as students get older."
1. Identify the sampling problems. Who is likely overrepresented or missing, and can the 65% be treated as an estimate for all 1,400 students?
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This is a self-selected convenience sample, not a random sample, so 65% describes the 186 people who chose to answer and nothing more.
The cafeteria table reaches only students in one of three lunch waves, and only those who buy lunch on campus and choose to stop at the table. The Instagram link reaches only students who follow the account and see the post in time. Both routes systematically miss students who are absent, who eat during another wave, who leave campus, or who are not on social media. Self-selection also means students with strong opinions are overrepresented; the 121 yes votes may reflect enthusiasm rather than the school.
Response rate is 186 ÷ 1,400 ≈ 13%, and even a genuinely random sample of 186 would carry a margin of error near ±7 percentage points. A defensible estimate would require randomly selecting students within each grade and following up with nonresponders.
2. Which measure of center should be reported for sleep hours, and how should the 14 low reporters be handled?
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Report the median, 7.2 hours. The mean of 6.9 is pulled down by a small, genuinely atypical group, so the distribution is skewed left, and with skew the median describes the typical student better than the mean.
The 14 students reporting 4 hours or less should not be deleted. They are a real subgroup - athletes with 5:45 a.m. practice - and their sleep is arguably the most relevant data in a debate about start times. Removing them would make the mean look better and hide the very problem being discussed.
Best practice is to report the median as the typical value, mention the mean and the direction of skew, and describe the low-sleep group separately with its size and cause, rather than blending it into one average.
3. Evaluate the claim that support "drops steadily as students get older," using the grade breakdown. Include the difference between the two conditional probabilities involving 12th graders.
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Compute each grade's support rate:
- 9th: 58/88 ≈ 65.9%
- 10th: 41/62 ≈ 66.1%
- 11th: 17/26 ≈ 65.4%
- 12th: 5/10 = 50.0%
That is flat across grades 9–11 and a drop only in 12th grade, where the sample is 10 students - two or three responses would move the rate by 20–30 points. There is no steady decline.
The two conditionals are also being confused. P(support | 12th grade) = 5/10 = 50%, but P(12th grade | support) = 5/121 ≈ 4.1%. The second is what the overall 65% reflects: seniors are only 10 of 186 respondents, so even a large difference among them barely moves the total, and the overall figure stays close to the 9–11 rates.
The claim should be dropped. A defensible sentence is: "Support was similar in grades 9–11 and lower among the ten seniors who responded, too few to draw a conclusion."