Mathematics • Year 10 • Unit 4 • Lesson 2
Collecting Data — Mixed Challenge
Pull together every idea from Lesson 2: the four sampling methods, the population-vs-sample distinction, and the warning that sample size alone never removes bias. Spot a Year 10 mistake, then design your own stratified sampling plan.
1. Mixed problems — choose the right tool
Each question targets a different idea from Lesson 2. Decide whether it is about naming a method, identifying bias, or computing stratified numbers before you start writing. 3 marks each
1.1 Name the sampling method for each: (a) every 50th car passing a checkpoint, (b) drawing 30 student numbers from a random-number generator, (c) handing out flyers to the first 20 students you see in the playground.
1.2 A school of 500 students has 200 girls and 300 boys. Calculate the proportional stratified sample sizes if the SRC wants a total sample of 50.
1.3 A poll surveys only people who answer a landline phone between 9 am and 11 am on a Tuesday. State which two groups are systematically excluded from the sample, and name the type of bias this creates.
1.4 A researcher uses random sampling but only manages a sample of 25 people from a population of 5,000. A friend uses convenience sampling with 800 people. Which sample is likely to be less biased, and why?
1.5 Define the difference between a census and a sample survey in your own words, and give one Australian example of each.
1.6 A small grocery chain has 8 stores. To survey customer satisfaction, the head office randomly chooses 2 of the 8 stores and surveys every customer who shops at those 2 stores during a week. (a) Name this sampling design. (b) Give one strength and one weakness of this design.
2. Find the mistake
Another Year 10 student has tried to design a stratified sample of a 600-student school and to explain why it is unbiased. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — School: 600 students = 150 Y7 + 150 Y8 + 100 Y9 + 100 Y10 + 50 Y11 + 50 Y12. Take a stratified sample of 60.
Line 1: Sample fraction = 60 ÷ 600 = 1/10.
Line 2: Take 10 students from every year group (60 ÷ 6 = 10 per year).
Line 3: This is stratified because each year group is represented in the sample.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working, including the corrected sample sizes for each year group.
Stuck? In proportional stratified sampling the sample size in each stratum must match its proportion of the population — not be equal across strata.3. Open-ended challenge — design a sampling plan
This question has many valid answers. Be creative but follow every constraint. 4 marks
3.1 Choose a research question of your own that affects students at your school. Then write a sampling plan that:
- names the population in one sentence,
- uses a stratified random sample with at least two strata (e.g. junior/senior, or by year level, or by class),
- has a total sample size between 50 and 100,
- shows the calculation for how many people are sampled from each stratum (so the proportions match the population),
- describes how randomness is achieved within each stratum (e.g. names in a hat, random-number generator),
- and finishes with one sentence explaining why your plan is less biased than asking the first 60 students you bump into.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Name the method
(a) Every 50th car → systematic sampling.
(b) Random-number generator on student numbers → simple random sampling.
(c) First 20 in the playground → convenience sampling.
1.2 — 50 from 500 (girls/boys)
Sample fraction = 50 ÷ 500 = 1/10. Girls: 200 × 1/10 = 20. Boys: 300 × 1/10 = 30. Total = 50 ✓.
1.3 — Landline 9-11 am Tuesday
Systematically excluded: (i) people who work weekday daytime hours and (ii) people who do not own a landline (most younger Australians use mobile only). This produces a strong selection bias (closely related to coverage bias) toward older, retired, or stay-at-home respondents.
1.4 — Small random vs large convenience
The random sample of 25 is likely less biased than the convenience sample of 800. Bias depends on the method, not the size — Lesson 2 is explicit: "Sample size alone does not eliminate bias." A small random sample may have wider variability, but it does not systematically over- or under-represent any group.
1.5 — Census vs sample survey
A census collects data from every member of the population. A sample survey collects data from a subset (sample) chosen to represent the population.
Example of census: the ABS Census of Population and Housing.
Example of sample survey: the ABS Monthly Labour Force Survey (about 26,000 households representing all of Australia).
1.6 — Cluster sampling
(a) Cluster sampling (randomly choose whole clusters — stores — then survey every member of those clusters).
(b) Strength: cheap and easy to run, since you only visit 2 stores instead of all 8. Weakness: results may be biased if the 2 chosen stores are not typical of the chain (e.g. both happen to be in the same suburb).
2 — Find the mistake
(a) The mistake is on Line 2.
(b) "10 per year" is equal, not proportional. A proportional stratified sample must match each stratum's share of the population — Y7 and Y8 each have 150 students, so they should contribute more to the sample than Y11 and Y12 (50 each).
(c) Corrected, using sample fraction 1/10: Y7: 15, Y8: 15, Y9: 10, Y10: 10, Y11: 5, Y12: 5. Check total: 15 + 15 + 10 + 10 + 5 + 5 = 60 ✓.
3 — Open-ended challenge (sample solution)
Research question: What proportion of students at our school have used ChatGPT for homework this term?
Population: All 800 students currently enrolled at Riverbend High, with 500 in Years 7-10 (juniors) and 300 in Years 11-12 (seniors).
Sample size & strata: Take a total stratified random sample of 80 (sample fraction = 80 ÷ 800 = 1/10). Juniors: 500 × 1/10 = 50; Seniors: 300 × 1/10 = 30. Check: 50 + 30 = 80 ✓.
How randomness is achieved: For each stratum, export the roll-number list, use a random-number generator (e.g. =RAND() in a spreadsheet) to sort, then take the top 50 (juniors) or top 30 (seniors).
Why less biased: Every junior and every senior has an equal chance of being chosen within their stratum, and the strata are represented in the same proportions as in the school — unlike a convenience sample of "the first 60 students I see", which would over-represent the people I personally cross paths with.
Marking: 1 mark for stating population, 1 for valid stratified split with correct proportional arithmetic, 1 for a real method of randomisation, 1 for an explicit comparison to a convenience approach. Any valid plan that meets every bullet earns full marks.