Mathematics Standard • Year 11 • Module 4 • Lesson 2
Collecting Data — Skill Drill
Build fluency in identifying census vs sample, naming the five sampling methods, and spotting the four common types of bias.
1. Quick recall
Answer each question in the space provided. 1 mark each
Q1.1 A census collects data from ____________ member of a population. A sample collects data from a ____________ of the population.
Q1.2 Name each sampling method from its definition:
"Every member of the population has an equal chance of being selected" → ______________________
"Population is split into groups, then sampled in proportion from each group" → ______________________
"Select every nth person from an ordered list" → ______________________
Q1.3 Name the four types of bias covered in the lesson: ____________, ____________, ____________, ____________.
2. Worked example — stratified sampling at a NSW high school
Follow each line of working. Every step has a reason on the right.
Problem. A high school of 500 Year 11 students (300 female, 200 male) wants to survey 50 students about homework time. Design a stratified sample.
Step 1 — Find each stratum's proportion of the population.
Female: 300 / 500 = 0.60 Male: 200 / 500 = 0.40
Reason: stratified sampling must keep each group's share of the sample equal to its share of the population.
Step 2 — Multiply each proportion by the desired sample size 50.
Females in sample = 0.60 × 50 = 30
Males in sample = 0.40 × 50 = 20
Reason: this distributes the 50 sample slots in proportion to the population split.
Step 3 — Apply simple random sampling inside each stratum.
Pick 30 names at random from the 300 female roster; pick 20 names at random from the 200 male roster.
Reason: randomness inside each stratum prevents selection bias within the group.
Conclusion. The stratified sample contains 30 females and 20 males, mirroring the 60/40 gender split of the school.
3. Faded example — fill in the missing steps
A NSW school has 800 students: 320 in Year 7, 280 in Year 8, 200 in Year 9. The principal wants a stratified random sample of 80 for a canteen survey. Calculate how many students from each year. 4 marks
Step 1 — Year 7 proportion: ____ / 800 = ____
Step 2 — Year 8 proportion: ____ / 800 = ____
Step 3 — Year 9 proportion: ____ / 800 = ____
Step 4 — Multiply each proportion by 80:
Year 7 sample = ____ Year 8 sample = ____ Year 9 sample = ____
Conclusion. Pick ____ Year 7s, ____ Year 8s, and ____ Year 9s at random from each year group. Check: total = ____ (should equal 80).
4. Graduated practice — sampling and bias
Answer in the space below each part. Use full sentences for the bias questions.
Foundation — single-step classification (4 questions)
| Q | Scenario | Census or sample? |
|---|---|---|
| 4.1 1 | The five-yearly Australian Bureau of Statistics population count. | |
| 4.2 1 | A teacher asks 30 of her 120 students how much homework they did. | |
| 4.3 1 | A school principal collects an attendance roll from every student each morning. | |
| 4.4 1 | A polling company calls 1,200 voters before a NSW state election. |
Standard — typical HSC difficulty (6 questions)
Name the sampling method or identify the bias, and give a one-sentence reason.
4.5 A researcher selects every 20th name from an alphabetical school roll of 600 students. Name the sampling method. 2 marks
4.6 A TV channel asks viewers to text in to vote on a political question, charging $1.10 per text. Identify two sources of bias. 2 marks
4.7 A council surveys "support for new bike lanes" by asking people leaving a cycling shop. Name the bias most clearly present. 2 marks
4.8 A school has 60% day students and 40% boarders. A stratified sample of 50 is needed. How many from each? 2 marks
4.9 A questionnaire asks "Don't you agree that the new uniform policy is too strict?" Name the type of bias this question creates and explain in one sentence. 2 marks
4.10 A company mails a satisfaction survey to 5,000 customers; 280 reply. Name the type of bias likely to dominate the results and explain in one sentence. 2 marks
Extension — design and reasoning (2 questions)
4.11 A council wants to survey commuters' satisfaction with a new train line. They have access to: (a) Opal tap-on records, (b) station exit interviews, (c) an online opt-in form on the transport website. Recommend ONE method, explain in one sentence why you chose it, and name one bias your method will still need to manage. 3 marks
4.12 A school of 1,200 students has populations: 400 Year 11s, 350 Year 10s, 250 Year 9s, 200 Year 8s. Design a stratified random sample of 60 students and explain in ONE sentence why stratification is better than a simple random sample of 60 for this purpose. 3 marks
5. Self-check the easy 3
Tick the first three once you've checked your method works.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1 — Census vs sample
A census collects from every member of a population. A sample collects from a subset.
Q1.2 — Naming sampling methods
Equal chance of selection → simple random sample.
Proportional from groups → stratified sample.
Every nth from a list → systematic sample.
Q1.3 — Four types of bias
Selection bias, response bias, non-response bias, measurement bias.
Q3 — Faded example (canteen survey)
Year 7: 320/800 = 0.40 → 0.40 × 80 = 32.
Year 8: 280/800 = 0.35 → 0.35 × 80 = 28.
Year 9: 200/800 = 0.25 → 0.25 × 80 = 20.
Total: 32 + 28 + 20 = 80. ✓
Q4.1 – Q4.4 — Census vs sample
4.1 Census (ABS counts every household). 4.2 Sample (30 of 120). 4.3 Census (every student). 4.4 Sample (1,200 of millions of voters).
Q4.5 — Every 20th name from a roll
Systematic sample. Reason: selecting every nth member from an ordered list is the definition of a systematic sample.
Q4.6 — Pay-to-text TV vote
Two sources: (i) self-selection bias — only people motivated enough to text vote, who tend to hold the strongest opinions; (ii) selection bias — the $1.10 cost excludes those unable or unwilling to pay. (Either of these scoring 1 mark; both for 2 marks.)
Q4.7 — Bike-lane survey outside cycling shop
Selection bias. The sample is restricted to people already interested in cycling, who will tend to support bike lanes — so the result will not generalise to the wider community.
Q4.8 — Day students vs boarders, stratified sample of 50
Day: 0.60 × 50 = 30. Boarders: 0.40 × 50 = 20. Total = 50. ✓
Q4.9 — Leading question on uniform
Response bias. The question is leading ("Don't you agree…") — it pushes respondents toward agreement, which distorts the data away from their true opinion.
Q4.10 — Low response rate to satisfaction survey
Non-response bias. The 280 who replied are unlikely to represent the 4,720 who did not — usually only the very satisfied or very dissatisfied take the time to respond, exaggerating both extremes.
Q4.11 — Train-line commuter survey
Recommend (b) station exit interviews sampled at multiple stations and time blocks. Reason: this reaches actual commuters at the point of use and is much less self-selected than (c), and unlike (a) it captures opinions rather than just travel patterns. Bias still to manage: time-of-day selection bias — peak-hour commuters differ from off-peak, so the sampling schedule must cover both. (Other reasoned answers are accepted.)
Q4.12 — Stratified sample of 60
Year 11: 400/1200 × 60 = 20. Year 10: 350/1200 × 60 = 17.5 → 17 or 18 (round to keep total = 60, so 17). Year 9: 250/1200 × 60 = 12.5 → 13. Year 8: 200/1200 × 60 = 10. Check: 20 + 17 + 13 + 10 = 60. ✓
Why stratified beats simple random: a simple random sample of 60 from 1,200 could under-represent smaller year groups by chance; stratification guarantees each year group's voice is heard in proportion.