8.10 Sampling

**Stratified sampling.** A school has 200 pupils in Year 7, 180 in Year 8, and 220 in Year 9. A survey of 90 pupils is to be taken using stratified random sampling. (a) How many from each year? (b) Compare with simple random sampling of 90 from the whole school. Which is more likely to over- or under-represent a year? (c) Suppose 20 new Year 7 pupils are added and the sample size stays 90. How does the Year-7 sample size change?

**Sampling bias scenarios.** For each scenario, state whether the sample is likely biased and explain. (a) Surveying mass transport use only by interviewing people on a tram. (b) Estimating average height from a basketball team. (c) Phone polling people about TV viewing habits at 9 am on a weekday.

**Predicting from a sample.** A sample of 25 Year 8 pupils contains 3 left-handers. The Y8 cohort has 400 pupils. (a) Estimate the number of left-handed Y8 pupils. (b) Identify one assumption being made. (c) The school's overall stat is "about 10% of pupils are left-handed". Does your estimate agree? Discuss.

**Designing a survey.** You want to know how many Y8 pupils enjoy maths. Design a survey: (a) Write three good survey questions. (b) Specify a sample size and sampling method. (c) State two ways to reduce bias.

**Sample size and confidence.** Two polls predict election support: - Poll A: 60% support, sample size 100, MoE ±10%. - Poll B: 60% support, sample size 1000, MoE ±3%. (a) Which poll is more reliable? Why? (b) For Poll A, the true support could be anywhere from ? to ? (c) For Poll B, ditto. (d) The candidate needs > 50% to win. Which poll is more conclusive?

**Stratified vs simple random.** A school has the following enrolment: | Year | 7 | 8 | 9 | 10 | 11 | |------|---|---|---|----|----| | Count | 150 | 140 | 130 | 120 | 100 | A sample of 64 is required, stratified by year. (a) Compute the sample size for each year (round to nearest integer). (b) The school decides to oversample Y11 to ensure ≥ 15 are in the sample. Revise.

**Bias detection.** Identify the type of bias in each scenario and suggest a fix. (a) Asking "Don't you agree that the maths curriculum is great?" (b) Calling phone numbers only during work hours. (c) Surveying only students at the after-school club about extracurriculars.

**Cleaning data.** A spreadsheet of student heights (cm) includes: 152, 148, 1500, NA, 153, 155, "?", 149, 151, 1.6. (a) Identify problematic entries. (b) Suggest a cleaning plan. (c) Compute the mean before and after cleaning.

**Sample-size planning.** A school of 800 students wants to estimate the percentage who walk to school, to within ±5%. (a) Estimate the required sample size (rule of thumb: $n \approx 1/p^2$ where $p$ is the desired margin in decimal form). (b) Compare with sampling 30 students. What is the approximate margin of error?

**Non-response bias.** A survey is sent to 500 students, but only 200 respond. (a) Compute the response rate. (b) Why is non-response a problem for estimates? (c) Suggest two ways to reduce non-response.

**Polling election support.** A poll surveys 1000 voters: 540 support A, 460 support B. The election needs >50% to win. (a) Find the sample percentages. (b) State the margin of error (rule of thumb). (c) Can the poll predict the winner with confidence?

**Question design audit.** A school survey asks: "How often do you study at home?" (a) Suggest 5 mutually exclusive, exhaustive response options. (b) Critique the question — what assumptions does it make? (c) Rewrite the question to remove the assumption.