Later start survey: a headline that does not add up
The student council at a secondary school of 1,150 students wants a 9:30 am start every Wednesday. They survey 240 students during one lunchtime in the canteen. Of the 240, 138 say they support the change. The council records year levels too: 190 of the respondents are Year 13, and 124 of those Year 13s support the change. Of the 50 respondents who are not Year 13, only 14 support it. The council plans to tell the board that '58% of students support a later start'. A teacher notes that Year 13 students make up only about 17% of the roll and that they tend to gather in the canteen at lunch. The council also plans to pick one respondent's name at random from the 240 for a $20 voucher.
1. Calculate the overall percentage supporting the change and the percentage in each year-level group. Explain why the 58% headline may mislead the board, using a weighted estimate based on the roll.
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Overall: 138 ÷ 240 = 0.575 = 57.5%, which rounds to the 58% headline.
Year 13: 124 ÷ 190 = 0.653 = 65.3% support. Non-Year 13: 14 ÷ 50 = 0.28 = 28% support.
The sample is not representative: 190 ÷ 240 = 79.2% of respondents are Year 13, but Year 13s are only about 17% of the roll. Using the roll proportions, a representative sample would be about 17% Year 13 and 83% other. Weighted estimate = 0.17 × 65.3% + 0.83 × 28% = 11.1% + 23.2% = 34.3%. So the headline overstates support substantially. The 58% figure describes the sample, not the school.
2. For the voucher draw, calculate: P(Year 13), P(supporter), P(Year 13 and supporter), and P(Year 13 | supporter). Explain what these probabilities do and do not show about the school.
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P(voucher goes to a Year 13) = 190 ÷ 240 = 0.792 (79.2%).
P(respondent supports) = 138 ÷ 240 = 0.575 (57.5%).
P(Year 13 and supports) = 124 ÷ 240 = 0.517 (51.7%).
P(Year 13 | supports) = 124 ÷ 138 = 0.899 (89.9%).
The last figure says that most supporters in the sample were Year 13, but that is largely because Year 13s were over-represented. It is not evidence that most students in the school support the change, nor that a random student who supports it is likely to be Year 13 in the whole school. Conditional probability must be read against how the sample was chosen.
3. How many Year 13s should a representative sample of 240 include? Using the support rates found in each group, estimate the number of supporters in such a sample. Suggest one practical improvement to the survey.
Read the worked answer
A representative sample of 240 should contain about 0.17 × 240 = 40.8, so about 41 Year 13s and 199 other students.
Using the support rates found in each group: expected supporters ≈ 41 × 0.653 + 199 × 0.28 = 26.8 + 55.7 = 82.5, so about 82 or 83 supporters out of 240, or 34–35%. That is very different from 138 out of 240.
The council should use a random sample from the roll, for example a stratified sample with 41 Year 13s and 199 students from other year levels, and survey at different times and places rather than only the canteen. They should report the sample size and the method, and avoid presenting a location-biased group as the whole school.