The School Canteen Survey
A secondary school with 840 students wants to know how often they buy lunch from the canteen. A prefect proposes standing at the canteen door on Monday at 12:30 and asking the first 60 students who come out. A teacher suggests instead drawing 60 names at random from the full register and interviewing them during form time. The head of year worries that students who never use the canteen are exactly the ones the survey needs to hear from, and notes that Monday is the day the canteen offers a discount. The prefect argues that their method is quicker and that anyone leaving the canteen is clearly a real customer, so their answers will be more accurate.
1. Identify the sampling method the prefect proposes and explain two distinct reasons why it may produce a biased estimate of canteen use across the whole school.
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The prefect proposes a convenience (opportunity) sample - specifically a quota of the first 60 people available at one location and time.
Reason 1: coverage bias. Only students who are at the canteen on Monday lunchtime can be selected. Students who bring packed lunches, eat elsewhere, or skip lunch have zero chance of selection, yet they are part of the population whose behaviour is being estimated. The sample systematically over-represents frequent canteen users.
Reason 2: timing bias. Monday is a discount day, so canteen attendance that day is probably higher than the weekly average. Even among canteen users, the sample reflects an unusually busy day, inflating any estimate of typical use.
A third point worth noting: the first 60 out of the door may over-represent students who leave early (perhaps younger or less supervised students), so even within canteen users the sample is not random.
2. The teacher's proposal is a simple random sample of 60 from the register. Explain why this removes the coverage problem but does not guarantee a precise estimate, and state one practical difficulty with carrying it out.
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A simple random sample gives every one of the 840 students a known, equal chance of selection, so non-canteen users can appear in the sample in roughly their true proportion. This removes the structural coverage bias.
However, randomness does not guarantee precision. Any single sample of 60 will differ from the population; the estimate carries sampling error. With a binary outcome (uses canteen / does not), the standard error of a proportion is about sqrt(p(1−p)/n). If p ≈ 0.6, that is sqrt(0.24/60) ≈ 0.063, so a 95% interval is roughly ±12 percentage points. That is wide, and it is wide regardless of how carefully names were drawn.
Practical difficulty: selected students may be absent, refuse, or be hard to locate during form time, and replacing them with whoever is available reintroduces the very bias the method was meant to avoid. Response rates must be tracked and non-response reported.
3. The head of year proposes stratifying by year group. Explain how this would be done and one condition under which it would improve on the simple random sample.
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Procedure: divide the register into strata by year group (e.g. Years 7–11), then draw a random sample within each stratum. To keep the overall sample at 60, allocate proportionally to stratum size - if Year 7 has 180 of 840 students, it contributes 60 × 180/840 ≈ 13 students. Sampling within each stratum must still be random.
When it helps: stratification reduces the standard error when canteen use differs substantially between year groups - for example, if older students leave school at lunch while younger ones must stay. In that case much of the variability is between strata, and removing it from the within-stratum comparison gives a more precise overall estimate than a simple random sample of the same size.
If year groups hardly differ in canteen use, stratification gains little and adds administrative cost.