Mathematics • Year 9 • Unit 4 • Lesson 10
Data Collection — Mixed Challenge
Pull together every idea from Lesson 10: census vs sample, simple random / stratified / systematic / convenience sampling, and the three types of bias. You'll choose the right tool for each problem, spot a flawed analysis, and design your own survey from scratch.
1. Mixed problems — choose the right tool
Each question uses a different idea from Lesson 10. Decide what method or bias applies before you start writing. Show your working. 3 marks each
1.1 A school has $1200$ students across years 7-12 (each year has the same number). Design a stratified random sample of $120$ students. Show year-by-year numbers and the total.
1.2 For each scenario, name the most likely sampling method AND the main type of bias (if any): (a) interviewing every $50$th customer leaving a store; (b) asking only the customers who use the store loyalty card; (c) phoning $300$ randomly-selected landline numbers; (d) using the electoral roll to randomly select $500$ adults.
1.3 An online petition asking "Should drink prices at events be capped?" gets $50\,000$ "Yes" votes. The petition organiser claims "$98\%$ of Australians support price caps." Identify the two main flaws in this claim using terms from Lesson 10.
1.4 A factory has $500$ workers: $300$ assembly, $150$ packaging, $50$ management. They want a stratified sample of $40$. Calculate the sample for each role, rounding to whole numbers, and check the total.
1.5 Explain in two sentences why the Australian Census (conducted every $5$ years) collects from every household, while the monthly Labour Force Survey uses a sample of about $50\,000$ — i.e. why a census is suitable in one case but not the other.
1.6 Rewrite the following biased survey questions to make them neutral (one sentence each):
(a) "Don't you agree that this school's lunch break is too short?"
(b) "Should taxpayer money be wasted on a new sports oval?"
(c) "Wouldn't you rather have more screen time at school?"
2. Find the mistake
Another student has tried to design a stratified sample. 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
Problem: A workplace has $400$ employees: $240$ in Sales, $120$ in Engineering, $40$ in HR. Design a stratified sample of $40$.
Line 1: Total $= 240 + 120 + 40 = 400$ ✓
Line 2: Sales proportion $= 240/400 = 0.6$; Eng $= 120/400 = 0.3$; HR $= 40/400 = 0.1$.
Line 3: Sample: Sales $= 0.6 \times 40 = 24$; Eng $= 0.3 \times 40 = 12$; HR $= 0.1 \times 4 = 0.4 \approx 0$.
Line 4: Total sample $= 24 + 12 + 0 = 36$.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected final sample numbers and total.
Stuck? Check the multiplication in Line 3 — there's a typo. Also: it's bad practice in stratified sampling to drop an entire stratum to $0$.3. Open-ended challenge — design your own survey
This question has many valid answers. 4 marks
3.1 You have been asked by your school principal to find out how the school's $900$ students would rate the new canteen menu on a scale of $1$-$5$.
Design a complete survey. Your design must include:
(i) The sampling method you will use, with at least one stratum named.
(ii) The sample size, with the proportion of the population it represents.
(iii) One neutral, non-leading survey question (don't ask "Isn't the new menu great?" — ask a balanced version).
(iv) One step you will take to reduce non-response bias (e.g. in-person follow-up, multiple reminders, an incentive).
(v) A brief justification (two sentences) of why your design will produce more reliable results than just asking the kids in your form class.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — School stratified sample 1200 → 120
Six year groups, each with $200$ students ($1200/6$). Each stratum's proportion is $\tfrac{1}{6}$, so sample per year $= \tfrac{1}{6} \times 120 = \mathbf{20}$ students.
Total: $6 \times 20 = \mathbf{120}$ ✓.
1.2 — Identify method + bias
(a) Systematic — generally low bias if customer arrival is random.
(b) Convenience sampling → selection bias (loyalty members aren't representative of all customers).
(c) Simple random — but mild selection bias: landline-only households skew older / more rural.
(d) Simple random from a high-quality frame — minimal bias (electoral roll covers all eligible adults).
1.3 — Online petition flaws
Two flaws: (1) Self-selection bias — only people who feel strongly about price caps sign petitions; they are not a representative sample of Australians. (2) The "$98\%$" figure has no denominator from the general population — the petitioner is comparing yes-votes to the (presumably small) number of "no" votes among petition signers, not to Australia's $\sim 18$ million voters. Without a known sampling frame, the percentage is meaningless.
1.4 — Factory stratified sample 500 → 40
Assembly: $\dfrac{300}{500} \times 40 = \mathbf{24}$. Packaging: $\dfrac{150}{500} \times 40 = \mathbf{12}$. Management: $\dfrac{50}{500} \times 40 = \mathbf{4}$.
Check: $24 + 12 + 4 = \mathbf{40}$ ✓.
1.5 — Census vs sample
The 5-yearly Census collects detailed long-term planning data (e.g. local population numbers needed to plan schools and hospitals in every suburb) where total accuracy matters, so a census is worth its high cost every $5$ years. The monthly Labour Force Survey needs to track a single fast-changing number (the unemployment rate) in time for monthly economic decisions, so a representative sample of $50\,000$ is more efficient than trying to census $20$ million adults each month.
1.6 — Rewrite biased questions
(a) "How would you rate the length of the school lunch break: too short / about right / too long?"
(b) "Do you support, oppose, or have no opinion on the proposed new sports oval?"
(c) "Compared with the current amount, how much screen time would you prefer at school: less / the same / more?"
The pattern: remove loaded words, offer balanced options, avoid "Don't you agree..." phrasing.
2 — Find the mistake
(a) The mistake is on Line 3.
(b) The HR calculation has a typo: it should be $0.1 \times \mathbf{40}$, not $0.1 \times 4$. With the typo, HR's sample of $0$ also breaks the principle of stratified sampling, which requires every stratum to be represented proportionally.
(c) Corrected working:
Sample: Sales $= 0.6 \times 40 = \mathbf{24}$; Eng $= 0.3 \times 40 = \mathbf{12}$; HR $= 0.1 \times 40 = \mathbf{4}$.
Total $= 24 + 12 + 4 = \mathbf{40}$ ✓.
3 — Open-ended challenge (sample design)
(i) Method: Stratified random sampling, stratified by year level (Years 7-12 = six strata).
(ii) Sample size: $90$ students total ($\dfrac{90}{900} = 10\%$). $15$ randomly selected students per year group.
(iii) Survey question (neutral, $1$-$5$ scale): "On a scale from 1 (very poor) to 5 (excellent), how would you rate the new canteen menu overall?" Use the same wording for every respondent and offer all five options on the same line.
(iv) Reducing non-response bias: Hand the survey out in person during a form class with $5$ minutes' completion time, so that response rate is close to $100\%$ — students cannot just ignore an email. Have the form teacher chase any who were absent.
(v) Justification: Asking just your form class would give a convenience sample heavily biased by who happens to be in your class and might miss whole year groups. Stratifying by year and randomly selecting within each year ensures the sample reflects the whole school's population in a way that a single-class sample cannot.
Marking: 1 mark for parts (i)+(ii) (sensible method + size); 1 for (iii) (a genuinely neutral question); 1 for (iv) (a specific anti-non-response step); 1 for (v) (clear justification referring to representativeness).