Mathematics • Year 9 • Unit 4 • Lesson 10
Sampling in the Real World
Use sampling methods and bias-spotting in everyday Australian contexts: a federal election poll, ABS Labour Force Survey, a school cafeteria survey, a sporting club newsletter, and a TV ratings panel. Then explain your method in your own words.
1. Word problems
Each problem is a real-world sampling scenario from Lesson 10. Show your working — vague claims of "biased" without naming the type only earn half marks.
1.1 — Federal election poll. A polling company wants a stratified sample of $1500$ voters that reflects Australia's population by state. The (approximate) state populations as percentages are: NSW $32\%$, Vic $26\%$, Qld $20\%$, WA $11\%$, SA $7\%$, Tas $2\%$, NT $1\%$, ACT $1\%$.
(a) Calculate how many voters from each state should be in the sample.
(b) Check that the total is $1500$. 3 marks
1.2 — ABS Labour Force Survey. The Australian Bureau of Statistics surveys around $50\,000$ people every month to estimate the unemployment rate. Australia's total adult population is roughly $20\,000\,000$.
(a) Calculate the sampling fraction, expressing it as a simple fraction and as a percentage.
(b) Explain in one sentence why the ABS does a sample rather than a full census every month. 3 marks
1.3 — School cafeteria survey. A Year 9 student wants to know whether the school's $800$ students prefer the cafeteria's pizza or sushi. They ask only the $30$ students in their own form class. The result: $20$ say pizza, $10$ say sushi.
(a) Calculate the sample's "preference for pizza" as a percentage.
(b) Identify the sampling method and the main type of bias.
(c) Suggest a better method that uses stratified sampling. Briefly state what strata you would use. 3 marks
1.4 — Sports club newsletter. A local sports club emails its weekly newsletter to $2000$ members asking "Are you happy with the new game-day uniform?" Only $300$ members reply, and of those $250$ ($83\%$) are unhappy.
(a) Identify the type of bias most affecting these results.
(b) Explain in two sentences why the "$83\%$ unhappy" figure is probably not a fair reflection of the membership's true views.
(c) Suggest a way to reduce this bias. 3 marks
1.5 — TV ratings panel. An Australian TV ratings company installs measurement boxes in a stratified sample of $5000$ households out of about $10\,000\,000$ in the country. The strata are based on state, household size, and age of household head.
(a) Calculate the sampling fraction.
(b) Suggest one additional stratum that might improve the sample (and say why).
(c) If a TV show has a national rating of $12\%$ from this sample, approximately how many households does that represent in total? 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate runs an Instagram poll asking their $500$ followers "Do you think the school should add more vegan lunch options?" $80\%$ say yes. They tell the principal: "$80\%$ of the school wants more vegan options — you should act on this." In your own words, explain (i) at least two reasons this conclusion is statistically unreliable, (ii) which type(s) of bias are present, and (iii) what method would give a more trustworthy answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Federal election stratified sample
(a) NSW: $0.32 \times 1500 = \mathbf{480}$. Vic: $0.26 \times 1500 = \mathbf{390}$. Qld: $0.20 \times 1500 = \mathbf{300}$. WA: $0.11 \times 1500 = \mathbf{165}$. SA: $0.07 \times 1500 = \mathbf{105}$. Tas: $0.02 \times 1500 = \mathbf{30}$. NT: $0.01 \times 1500 = \mathbf{15}$. ACT: $0.01 \times 1500 = \mathbf{15}$.
(b) Total: $480 + 390 + 300 + 165 + 105 + 30 + 15 + 15 = \mathbf{1500}$ ✓.
This is exactly how real Australian election polling works — see the lesson's "Real-World Anchor" on Market Research and Polling.
1.2 — ABS Labour Force Survey
(a) Sampling fraction $= \dfrac{50\,000}{20\,000\,000} = \dfrac{1}{400} = \mathbf{0.25\%}$.
(b) A monthly census of every adult would be enormously expensive (tens of millions of dollars) and slow (would take longer than a month to collect and process), so the ABS uses a sample to produce monthly unemployment estimates quickly and cheaply.
1.3 — School cafeteria
(a) $\dfrac{20}{30} \times 100\% = \mathbf{66.7\%}$ prefer pizza.
(b) Convenience sampling → selection bias. The student's own form class may not reflect the whole school's preferences (e.g. all in one year level, may share friends with similar tastes).
(c) Stratify by year group (Years 7-12). Sample, say, $10$ students from each year's roll using random selection. Total $= 60$ students, representative across all year levels.
1.4 — Sports club newsletter
(a) Non-response bias (with strong self-selection element).
(b) Only $300$ of $2000$ members ($15\%$) responded. People with strong opinions — especially unhappy ones — are far more likely to bother replying than satisfied members, so the $83\%$ unhappy figure heavily overstates true dissatisfaction.
(c) Survey a smaller, randomly chosen subset of members in person or by phone with follow-ups to chase non-responders. Aim for $> 80\%$ response rate so the result reflects all members, not just the vocal ones.
1.5 — TV ratings panel
(a) $\dfrac{5000}{10\,000\,000} = \dfrac{1}{2000} = \mathbf{0.05\%}$.
(b) Sensible additional stratum: income bracket (or urban vs rural; presence of children; language spoken at home). Different income groups watch different programmes, so missing this stratum could under-represent some viewer segments.
(c) Estimated households $= 12\% \times 10\,000\,000 = \mathbf{1\,200\,000}$ households watching.
2.1 — Explain your thinking (sample response)
(i) The conclusion is unreliable for several reasons: (1) the sample is only the classmate's followers, not the whole school — followers self-selected to follow that account, which already biases who sees the poll; (2) the $80\%$ figure only reflects those who chose to vote — students who don't care or who disagree may not bother (self-selection / non-response bias); (3) Instagram followers are not necessarily current students of the school at all. (ii) Both selection bias (the audience is not the school population) and non-response / self-selection bias (only motivated voters participate). (iii) A more trustworthy method would be a stratified random sample drawn from the official school roll, perhaps $5$ students per year level, surveyed in person with $> 80\%$ response — that produces a sample that actually reflects the school's $800$ students.
Marking: 1 for each of at least two distinct reasons; 1 for naming the correct bias type(s); 1 for proposing stratified random sampling as the improvement.