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Solutions — Full Answer Key
MathematicsYear 8 · 8.10 Sampling
Solutions · Full Answer Key
Pack A answers · Pack B answers · Problem-solving worked solutions
Pack A — Answers
Bronze
1.Survey: a study collecting data from some or all of a group. Census: data from every member of a population. Population: the entire group of interest. Sample: a subset of the population studied.
2.Population: 600 students; Sample: 30 students
3.Simple random sampling
4.10%
5.60 students
6.Leading question — assumes the food is poor
7.520
8.Self-selection bias: library users read more than average
9.24%
10.Early arrivers are not representative — they may live near school or be more punctual
Silver
11.Year 7: 20, Year 8: 18, Year 9: 22
12.55% — moderately confident with n=200
13.Stratified random sample by class/gender, sample size ≥ 50; use anonymous questionnaire
14.(a) census; (b) systematic sample (random-ish); (c) biased
15.Not necessarily — small sample (n=30), one class only (may be selected by ability or homeroom).
16."How often do you eat the cafeteria food?" with options: never / 1-2 times/week / 3-4 / 5+
17.45% is much higher than 10% — sample likely not representative or class is special
18.Trip participants are self-selected (only those whose parents allow it, those who can afford it); often a single year-group.
19.(d) Stratified across year-groups
20.1500 is impossible (15 m) — likely a typo for 150 or 155. Remove or correct.
Gold
21.Y7: 20, Y8: 18, Y9: 22, Y10: 10, Y11: 10
22.Estimate ≈ 6.5 h; n = 30 is small with range 4 → moderate confidence
23. — practically impossible; full census is needed
24.(a) 65% (b) 390 (c) Margin ≈ 9% — could be 56–74% in the wider group
25.Forces a choice between only two options; what about no pets / neither / both equally? Improvement: "Which do you prefer: cats / dogs / both equally / neither?"
26.(i) is better — random; (ii) introduces selection bias (early arrivers).
27.Method A: random call list. Pro: random; Con: phone bias. Method B: fitness-app users. Pro: large data; Con: heavily biased (active users only).
28.Overlap: 10 is in both (c) and (b); the upper limit of (b) should be 9 or 9.99.
29.≈ 4.7 distinct names
30.Not particularly biased — Likert scale; balanced 5 options.
Platinum
31.Pure stratified gives Y7: 20, Y8: 20, Y9: 20, Y10: 10, Y11: 10. Both Y10 and Y11 already get 10, > 5. So sample sizes: 20, 20, 20, 10, 10.
32.Estimate mean ≈ 18 kg; high spread (range 30 kg → many breeds → mean less informative)
33.95% confident the true support is 48–54%. Cannot rule out A being below 50% (i.e. losing).
34.200/300 = 67% (among respondents); could overestimate if pet owners more likely to respond.
35.Selection (only gate-passers), non-response (refusers), social desirability (under-reporting). Reduce by random in-class survey + anonymity.
36.90/100 = 90%? No — recompute using stratified weights.
37.Sample = 1 class, not the school. Cannot generalise without further sampling.
38.95% confident the true population proportion lies in 30–40%. Does NOT mean any individual has 30–40% chance of liking pizza.
39."How many days per week do you exercise for at least 30 min? 0/1-2/3-4/5-6/7" — stratified random sample across years.
40.Investigate zeros (absences? actual zeros?). 100 is plausible (perfect score). Missing entries: contact source or exclude. Document changes.
Pack B — Answers
Bronze
1.Same definitions.
2.Population: 800; Sample: 25
3.Random sampling (drawing names)
4.8%
5.75 students
6.Leading question — assumes more sports lessons are good
7.420
8.Selection bias: football players favour football
9.30%
10.Bus users are biased — exclude walkers
Silver
11.Same.
12.50% — similar
13.Same logic
14.Same.
15.Same.
16."What is your view on phones in school? Options: Allow / Restrict / Ban."
17.8% is close to 10% — consistent
18.Same.
19.Same.
20.220°C impossible for room temperature; correct to 22.
Gold
21.Apply 1/10 to each.
22.Estimate 7.0 h; better confidence with n = 50
23.
24.(a) 65% (b) 520 (c) Margin ≈ 7%
25.Same.
26.Same.
27.Same.
28.Same.
29.Same.
30.Same.
Platinum
31.Same.
32.Same.
33.Same.
34.Same.
35.Same.
36.Same.
37.Same.
38.Same.
39.Same.
40.Same.
Problem-solving — Worked Solutions
1Problem 1
Answer
(a) Y7: 30, Y8: 27, Y9: 33 (b) Simple random may by chance over- or under-represent (c) Y7 sample rises from 30 to ~32
Full working
(a) Total 600. Fraction 90/600 = 3/20. Years: 30, 27, 33. Total 90 ✓.
(b) Simple random could (by chance) pick disproportionately many of any single year. Stratified guarantees the year-group proportions match the population.
(c) New total 620. Fraction . Year 7 (now 220): pupils.
(b) Simple random could (by chance) pick disproportionately many of any single year. Stratified guarantees the year-group proportions match the population.
(c) New total 620. Fraction . Year 7 (now 220): pupils.
2Problem 2
Answer
(a) Biased (b) Biased (c) Biased
Full working
(a) Tram users are over-represented; non-tram users excluded.
(b) Basketball selects for height — sample over-represents tall people.
(c) Time bias: working people unavailable at 9 am, so the sample skews towards retired/unemployed/work-from-home.
(b) Basketball selects for height — sample over-represents tall people.
(c) Time bias: working people unavailable at 9 am, so the sample skews towards retired/unemployed/work-from-home.
3Problem 3
Answer
(a) 48 (b) Sample representativeness (c) Estimate (12%) is close to 10% — within sampling error
Full working
(a) Proportion 3/25 = 0.12. Estimate .
(b) The sample is assumed to be representative of the year-group (random, unbiased).
(c) 12% vs 10% — close. With , sampling error is ±5–10%, so consistent.
(b) The sample is assumed to be representative of the year-group (random, unbiased).
(c) 12% vs 10% — close. With , sampling error is ±5–10%, so consistent.
4Problem 4
Answer
(a) Sample given in working (b) Stratified random, n = 60 (c) Anonymous; balanced response options
Full working
(a) Sample questions:
1. "How often do you enjoy maths lessons? (Always / Often / Sometimes / Rarely / Never)"
2. "What aspects of maths do you most enjoy? (Algebra / Geometry / Statistics / Number / None)"
3. "How would you rate maths compared to other subjects? (Much more enjoyable / More / Same / Less / Much less)"
(b) Stratified random sample of 60 across all Y8 classes (10 from each class).
(c) Anonymous responses to reduce social-desirability bias. Balanced response options to avoid leading questions.
1. "How often do you enjoy maths lessons? (Always / Often / Sometimes / Rarely / Never)"
2. "What aspects of maths do you most enjoy? (Algebra / Geometry / Statistics / Number / None)"
3. "How would you rate maths compared to other subjects? (Much more enjoyable / More / Same / Less / Much less)"
(b) Stratified random sample of 60 across all Y8 classes (10 from each class).
(c) Anonymous responses to reduce social-desirability bias. Balanced response options to avoid leading questions.
5Problem 5
Answer
(a) Poll B (larger sample → smaller MoE) (b) 50–70% (c) 57–63% (d) Poll B (CI doesn't cross 50%)
Full working
(a) Larger sample reduces sampling error. MoE ∝ .
(b) Poll A: 60% ± 10% = 50%–70%.
(c) Poll B: 60% ± 3% = 57%–63%.
(d) Poll B's interval (57–63%) lies entirely above 50% → the candidate is statistically certain to win at the 95% confidence level. Poll A's interval includes 50% — too uncertain.
(b) Poll A: 60% ± 10% = 50%–70%.
(c) Poll B: 60% ± 3% = 57%–63%.
(d) Poll B's interval (57–63%) lies entirely above 50% → the candidate is statistically certain to win at the 95% confidence level. Poll A's interval includes 50% — too uncertain.
6Problem 6
Answer
(a) Y7: 16, Y8: 15, Y9: 14, Y10: 13, Y11: 11 (total 69 — over by 5; reduce others) (b) Move pupils from larger years to Y11
Full working
(a) Total 640. Fraction 64/640 = 1/10. Sizes: 15, 14, 13, 12, 10. Total 64.
(b) To get Y11 to 15, add 5 to Y11 and reduce others by 5 in proportion: e.g. Y11: 15, Y10: 11, Y9: 12, Y8: 13, Y7: 13. Total 64. Adjust as needed.
(b) To get Y11 to 15, add 5 to Y11 and reduce others by 5 in proportion: e.g. Y11: 15, Y10: 11, Y9: 12, Y8: 13, Y7: 13. Total 64. Adjust as needed.
7Problem 7
Answer
(a) Leading wording (b) Time-of-day bias (c) Selection bias
Full working
(a) Loaded wording — replace with: "How would you rate the maths curriculum? (5-point scale)"
(b) Workers may be unavailable at work hours. Sample evenings/weekends too.
(c) Self-selection bias — club attendees aren't representative of the whole school. Sample non-club students too.
(b) Workers may be unavailable at work hours. Sample evenings/weekends too.
(c) Self-selection bias — club attendees aren't representative of the whole school. Sample non-club students too.
8Problem 8
Answer
(a) 1500, NA, "?", 1.6 are problematic (b) Correct or remove (c) Before: undefined or ≈ 207; after ≈ 151
Full working
(a) 1500 = typo (15 m impossible). NA and "?" = missing. 1.6 = unit error (likely 1.6 m = 160 cm).
(b) Strategy: (i) correct typos where clear (1500 → 150, 1.6 → 160). (ii) Remove or impute NA / "?".
(c) Before: any computation is misleading. After: replace problems → e.g. 152, 148, 150, 153, 155, 149, 151, 160. Mean = cm.
(b) Strategy: (i) correct typos where clear (1500 → 150, 1.6 → 160). (ii) Remove or impute NA / "?".
(c) Before: any computation is misleading. After: replace problems → e.g. 152, 148, 150, 153, 155, 149, 151, 160. Mean = cm.
9Problem 9
Answer
(a) ≈ 400 (b) ±18% — too imprecise
Full working
(a) Margin of error . Set : .
(b) For : → MoE ≈ ±18%. Far too imprecise to make any meaningful conclusion.
(b) For : → MoE ≈ ±18%. Far too imprecise to make any meaningful conclusion.
10Problem 10
Answer
(a) 40% (b) Non-respondents may differ from respondents (c) Reminders, incentives
Full working
(a) .
(b) Non-respondents may have different views from respondents (e.g. busy students may differ in attitudes). Estimates based on respondents only are biased.
(c) (i) Send reminders. (ii) Offer small incentives (e.g. raffle entry). (iii) Make the survey shorter / easier.
(b) Non-respondents may have different views from respondents (e.g. busy students may differ in attitudes). Estimates based on respondents only are biased.
(c) (i) Send reminders. (ii) Offer small incentives (e.g. raffle entry). (iii) Make the survey shorter / easier.
11Problem 11
Answer
(a) A 54%, B 46% (b) MoE ≈ ±3% (c) Yes — A's 54% is outside the 50% threshold + MoE
Full working
(a) 54% A, 46% B.
(b) → MoE ±3.2%.
(c) A: 54% ± 3.2% → 50.8%–57.2%. Entirely above 50% → A predicted winner with confidence.
(b) → MoE ±3.2%.
(c) A: 54% ± 3.2% → 50.8%–57.2%. Entirely above 50% → A predicted winner with confidence.
12Problem 12
Answer
(a) Never / 1-2 days/wk / 3-4 / 5-6 / Every day (b) Assumes study happens; might not (c) "Do you study at home? If yes, how often? (options)"
Full working
(a) Mutually exclusive and exhaustive: 0 days, 1-2 days, 3-4 days, 5-6 days, 7 days. Every value falls in exactly one bucket.
(b) Assumes everyone studies at home at some level — but some may study elsewhere or not at all. Forcing a frequency response misses these cases.
(c) Two-step question: "Do you study at home? (Yes / No)" → if Yes, "How often?" with the options above.
(b) Assumes everyone studies at home at some level — but some may study elsewhere or not at all. Forcing a frequency response misses these cases.
(c) Two-step question: "Do you study at home? (Yes / No)" → if Yes, "How often?" with the options above.