Mathematics Standard • Year 12 • Module 8 • Lesson 1
Types of Data and Sampling — Past-Paper Style
Practise HSC Mathematics Standard 2-style writing on data classification, sampling design and bias evaluation.
1. Short-answer questions
1.1 Classify each of the following variables as nominal categorical, ordinal categorical, discrete numerical, or continuous numerical: (a) postcode, (b) Olympic medal won (gold/silver/bronze), (c) number of goals scored in an AFL match, (d) running time of a 100m sprint. 2 marks Band 3
1.2 A school of 800 students has 480 boys and 320 girls. A stratified random sample of 50 students is to be selected.
(a) How many boys should be sampled?
(b) How many girls should be sampled?
(c) Explain why stratified sampling is more appropriate than simple random sampling in this context. 3 marks Band 3-4
1.3 A radio station phones a sample of listeners during its 9-10am breakfast show and asks "Do you support our brave council in opposing the new highway?" 78% of respondents say yes.
(a) Identify two distinct sources of bias in this poll.
(b) For each bias, state whether it would push the reported "yes" rate up or down. 4 marks Band 4
2. Extended response
2.1 A regional council in NSW wants to investigate whether residents support the introduction of paid weekend parking in the town centre. The council has the following data and design choices to consider.
Population: 15,000 adult residents across three zones — Town Centre (3,000), Inner Suburbs (7,000) and Outer Suburbs (5,000).
Budget allows: a survey of 300 residents.
Initial draft question: "Do you agree with the council's responsible plan to charge for weekend parking to fund local services?"
(a) Identify the most appropriate sampling method and calculate the number of residents to sample from each of the three zones. Justify your choice.
(b) Identify two specific weaknesses of the initial draft question and rewrite it in a neutral form.
(c) Suggest one practical step the council could take to reduce non-response bias, and explain in one sentence why non-response bias is a particular risk for this survey topic. 7 marks Band 5-6
Explicit marking criteria
Part (a) — 3 marks
• 1 mark — names stratified random sampling (or equivalent) and justifies it (zones likely differ in opinion).
• 1 mark — correct stratum proportions: 60, 140, 100.
• 1 mark — shows working (300 × zone share) and confirms total = 300.
Part (b) — 2 marks
• 1 mark — identifies two specific issues (e.g. loaded wording "responsible plan", framing "to fund local services").
• 1 mark — neutral rewrite that removes both loaded phrases and offers balanced response options.
Part (c) — 2 marks
• 1 mark — proposes a concrete step (e.g. follow-up reminders, multiple modes, paid incentive).
• 1 mark — explains why non-response bias would skew this topic (opposers more motivated, or supporters silent — either direction acceptable if reasoned).
Your response:
Stuck on (a)? Multiply 300 by each zone's share of 15,000.How did this worksheet feel?
What I'll revisit before next class:
1.1 — Four classifications (2 marks)
Sample response. (a) Postcode → nominal categorical (digits used as labels — adding two postcodes is meaningless). (b) Olympic medal → ordinal categorical (gold > silver > bronze). (c) Goals scored → discrete numerical (whole-number count). (d) Running time → continuous numerical (measured time).
Marking notes. 1 mark for any 2 correct; 2 marks for all 4 correct. Common loss: calling postcode "discrete numerical" — it isn't, because the digits do not have arithmetic meaning.
1.2 — Stratified sample of 50 (3 marks)
Sample response. (a) Boys: 50 × 480/800 = 50 × 0.6 = 30 boys. (b) Girls: 50 × 320/800 = 50 × 0.4 = 20 girls. (c) Stratification keeps the boy:girl ratio in the sample the same as in the school (60:40). Simple random sampling could, by chance, produce a sample of (say) 38 boys and 12 girls, leaving the smaller group under-represented and biasing the result on any gender-related question.
Marking notes. 1 mark each for parts (a) and (b) with working shown. 1 mark for part (c) with a complete justification (mentioning proportions and the risk of under-representation). A bare answer "stratified is better" earns 0 for (c).
1.3 — Biased radio poll (4 marks)
(a) Sample response. (1) Measurement (leading-question) bias — the words "brave" and the assumption-of-opposition pre-frame a "yes" answer. (2) Selection bias — only people listening to that station between 9-10am can answer; this excludes working commuters and non-listeners, who may hold different views.
(b) Sample response. Both biases push the "yes" rate upwards: the leading wording makes "yes" feel like the right answer, and the audience self-selects to fans of that station's editorial line (who tend to share its views).
Marking notes. 1 mark per distinct bias correctly named. 1 mark for each correct direction with a one-sentence reason. A response identifying both biases but giving no direction = 2/4. A response naming "bias" generically (with no type) = 0 even if direction is correct.
2.1 — Council parking survey (7 marks): Band-6 sample with annotations
(a) Sampling method and per-stratum sample sizes.
Use stratified random sampling with zones as strata, because opinion on weekend town-centre parking is likely to differ systematically between residents who live in the town centre (would pay), inner suburbs (visit often) and outer suburbs (visit less often). [1 mark — method named and justified.]
Town Centre: 300 × 3,000/15,000 = 300 × 0.20 = 60.
Inner Suburbs: 300 × 7,000/15,000 = 300 × 0.4667 = 140.
Outer Suburbs: 300 × 5,000/15,000 = 300 × 0.3333 = 100.
Check: 60 + 140 + 100 = 300. [1 mark — correct proportions. 1 mark — working shown and total verified.]
(b) Weaknesses + neutral rewrite.
Issues: (1) the word "responsible" frames opposition as irresponsible — measurement bias; (2) the phrase "to fund local services" attaches a positive consequence, biasing responders towards "yes". [1 mark — two specific issues named.]
Neutral rewrite: "Do you support the introduction of paid parking in the town centre on weekends? ◯ Yes ◯ No ◯ Unsure". [1 mark — both loaded phrases removed and balanced response options offered.]
(c) Reducing non-response bias.
Practical step: send two follow-up reminders to non-responders within 10 days, and offer the survey in multiple modes (online + telephone + paper drop-off at the library), to give equal access regardless of internet skill or workday availability. [1 mark — concrete step with detail.]
Why it matters here: paid-parking is the kind of topic where opponents are more motivated to respond than supporters (people who don't drive into town don't care), so without follow-ups the "no" rate would be over-stated. [1 mark — topic-specific reason for non-response bias.]
Total: 7/7.
Band descriptors for marker.
Band 3: Names stratified sampling and calculates at least one stratum correctly; vague answer for (b) and (c). ≈ 2-3 marks.
Band 4: All three stratum sample sizes correct; identifies bias in the draft question but rewrite is partial; vague non-response step. ≈ 4-5 marks.
Band 5: Full (a) and (b); part (c) gives a concrete step but the topic-specific reason is generic. ≈ 6 marks.
Band 6: Stratified design fully justified, balanced rewrite, concrete multi-mode follow-up, and a topic-specific argument linking non-response bias to the parking question. 7/7.