Answer Key
8.10 Sampling
Pack A — Answers
| # | Question | Answer |
|---|---|---|
| 1 | Distinguish: a (a) survey, (b) census, (c) population, (d) sample. Give one-sentence definitions. | 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 | A school has 600 students. A class of 30 is surveyed. Identify the population and the sample. | Population: 600 students; Sample: 30 students |
| 3 | A researcher gives each student a number and uses a random-number generator to pick 30. What sampling method is this? | Simple random sampling |
| 4 | A sample of 60 out of 600 pupils is taken. What percentage is the sample of the population? | 10% |
| 5 | In a sample of 25 students, 3 were left-handed. Estimate the number of left-handed students in a school of 500. | 60 students |
| 6 | A survey question reads "Do you agree that the canteen serves poor food?" Why is this biased? | Leading question — assumes the food is poor |
| 7 | A survey of 100 pet owners finds 65% have dogs. Estimate how many dog owners there are in a community of 800 pet owners. | 520 |
| 8 | A teacher surveys only students who attend the library to ask about reading habits. State one source of bias. | Self-selection bias: library users read more than average |
| 9 | In a sample of 50 cars, 12 are red. Estimate the percentage of red cars in the population. | 24% |
| 10 | A teacher takes the first 30 students arriving at school as a sample. Identify a potential bias. | Early arrivers are not representative — they may live near school or be more punctual |
| 11 | A school has Year 7: 200, Year 8: 180, Year 9: 220. A stratified sample of 60 is to be taken. How many from each year? | Year 7: 20, Year 8: 18, Year 9: 22 |
| 12 | A poll of 200 voters finds 110 support candidate A. Estimate the percentage of supporters in the wider population, and comment on how confident this estimate is. | 55% — moderately confident with n=200 |
| 13 | A researcher wants to learn about Year 8 lunch preferences. Suggest a sampling plan that avoids bias. | Stratified random sample by class/gender, sample size ≥ 50; use anonymous questionnaire |
| 14 | Identify whether each scenario is a census, a sample or biased: (a) UK government population census every 10 years. (b) Surveying every 10th student exiting the school. (c) Asking the football team about favourite sport. | (a) census; (b) systematic sample (random-ish); (c) biased |
| 15 | A class of 30 students is surveyed about study time. The mean is 7.4 hours/week. Is this representative of all Y8 students? Suggest two reasons it might not be. | Not necessarily — small sample (n=30), one class only (may be selected by ability or homeroom). |
| 16 | Improve this question: "How often do you eat the disgusting cafeteria food?" Provide a fairer version. | "How often do you eat the cafeteria food?" with options: never / 1-2 times/week / 3-4 / 5+ |
| 17 | A sample of 40 students has 18 left-handed. The expected proportion is 10%. Comment on whether the sample is consistent with this. | 45% is much higher than 10% — sample likely not representative or class is special |
| 18 | A teacher conducts a survey on a school trip. List two reasons why this sample is unlikely to represent the whole school. | Trip participants are self-selected (only those whose parents allow it, those who can afford it); often a single year-group. |
| 19 | Identify the most representative sampling method for a school survey: (a) volunteers in the library; (b) random class draw; (c) every 10th student exiting; (d) stratified across year-groups. | (d) Stratified across year-groups |
| 20 | A data set has the values 152, 148, 1500, 155 (heights in cm). Clean it and explain. | 1500 is impossible (15 m) — likely a typo for 150 or 155. Remove or correct. |
| 21 | A school survey on phone use has 800 students. A stratified random sample of 80 is taken: Year 7 (200), Year 8 (180), Year 9 (220), Year 10 (100), Year 11 (100). How many from each year? | Y7: 20, Y8: 18, Y9: 22, Y10: 10, Y11: 10 |
| 22 | A sample of 30 students reports an average of 6.5 h homework per week with range 4. Estimate the school-wide average (school of 600 students) and discuss the confidence in this estimate. | Estimate ≈ 6.5 h; n = 30 is small with range 4 → moderate confidence |
| 23 | For a survey about lunch preferences in a school of 1000 students, you want results accurate to ±3%. Approximately how large a sample is needed? | $n \approx 1067$ — practically impossible; full census is needed |
| 24 | A school surveys 120 students about whether they bring their own lunch. 78 say yes. (a) What percentage in the sample? (b) Estimate the count in a year-group of 600. (c) Comment on accuracy. | (a) 65% (b) 390 (c) Margin ≈ 9% — could be 56–74% in the wider group |
| 25 | A survey question is "Do you prefer cats over dogs?" State two ways this question is poorly designed and improve it. | 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 | A teacher considers two sampling plans for a school of 600. (i) 60 random students from the whole school; (ii) the first 60 students arriving at school. State which is better and why. | (i) is better — random; (ii) introduces selection bias (early arrivers). |
| 27 | A doctor wants to estimate the average daily steps for the local town (~50 000 people). Suggest two methods, with pros and cons. | 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 | A school survey contains the question: "How many hours of TV do you watch a week? (a) 0–5, (b) 6–10, (c) 10–15, (d) more than 15." Identify the problem with the response options. | Overlap: 10 is in both (c) and (b); the upper limit of (b) should be 9 or 9.99. |
| 29 | Sample with replacement vs without replacement. A class of 30 students has names placed in a hat. Five draws are made (with replacement). On average, how many distinct names are drawn? | ≈ 4.7 distinct names |
| 30 | In an opinion poll, respondents are asked: "How satisfied are you with the school cafeteria? Very satisfied / Satisfied / Neutral / Dissatisfied / Very dissatisfied." Identify whether this is biased. | Not particularly biased — Likert scale; balanced 5 options. |
| 31 | A school of 800 students will be surveyed about transport. Year 7: 200, Year 8: 200, Year 9: 200, Year 10: 100, Year 11: 100. A stratified sample of 80 is required, but you want to ensure ≥ 5 from Year 11. How many from each year? | 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 | A sample of 50 dogs at a vet has mean weight 18 kg, range 30 kg. The vet treats 1000 dogs/year. Estimate the population mean and discuss the spread. | Estimate mean ≈ 18 kg; high spread (range 30 kg → many breeds → mean less informative) |
| 33 | A polling firm surveys 1000 voters and reports candidate A at 51% support. Margin of error ±3%. What does this mean for confidence? | 95% confident the true support is 48–54%. Cannot rule out A being below 50% (i.e. losing). |
| 34 | A pet-owner survey has 60% response rate (out of 500 sent). 200 responded "yes, owns a pet". Estimate the percentage of pet owners in the surveyed population. | 200/300 = 67% (among respondents); could overestimate if pet owners more likely to respond. |
| 35 | A student conducts a survey at the school gate asking about phone use. Identify three types of bias possible and how to reduce each. | Selection (only gate-passers), non-response (refusers), social desirability (under-reporting). Reduce by random in-class survey + anonymity. |
| 36 | A stratified sample of 100 from a population of 1000 is split: 600 women and 400 men. After sampling, the survey finds 60 women say "yes" and 30 men. Estimate the overall population proportion of "yes" responses. | 90/100 = 90%? No — recompute using stratified weights. |
| 37 | A class of 28 students completes a maths test. The teacher's class mean is 16/20 with range 7. The teacher reports "the school average is 16/20". Critique this claim. | Sample = 1 class, not the school. Cannot generalise without further sampling. |
| 38 | A survey of 200 students yields a 95% confidence interval of "30–40% prefer pizza". Interpret this interval correctly. What does **not** it mean? | 95% confident the true population proportion lies in 30–40%. Does NOT mean any individual has 30–40% chance of liking pizza. |
| 39 | Design an unbiased question to estimate the proportion of students in a school who exercise daily, and state the sampling method. | "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 | A teacher cleans a dataset of 200 marks. She notices 15 zeros, 1 value of 100, and 5 missing entries. Outline a cleaning strategy. | Investigate zeros (absences? actual zeros?). 100 is plausible (perfect score). Missing entries: contact source or exclude. Document changes. |
Pack B — Answers
| # | Question | Answer |
|---|---|---|
| 1 | Distinguish: a (a) survey, (b) census, (c) population, (d) sample. Give one-sentence definitions. | Same definitions. |
| 2 | A school has 600 students. A class of 30 is surveyed. Identify the population and the sample. | Population: 800; Sample: 25 |
| 3 | A researcher gives each student a number and uses a random-number generator to pick 30. What sampling method is this? | Random sampling (drawing names) |
| 4 | A sample of 40 out of 500 pupils is taken. What percentage is the sample of the population? | 8% |
| 5 | In a sample of 40 students, 5 were left-handed. Estimate the number of left-handed students in a school of 600. | 75 students |
| 6 | A survey question reads "Do you agree that the canteen serves poor food?" Why is this biased? | Leading question — assumes more sports lessons are good |
| 7 | A survey of 100 pet owners finds 65% have dogs. Estimate how many dog owners there are in a community of 800 pet owners. | 420 |
| 8 | A teacher surveys only students who attend the library to ask about reading habits. State one source of bias. | Selection bias: football players favour football |
| 9 | In a sample of 50 cars, 12 are red. Estimate the percentage of red cars in the population. | 30% |
| 10 | A teacher takes the first 30 students arriving at school as a sample. Identify a potential bias. | Bus users are biased — exclude walkers |
| 11 | A school has Year 7: 200, Year 8: 180, Year 9: 220. A stratified sample of 60 is to be taken. How many from each year? | Same. |
| 12 | A poll of 200 voters finds 110 support candidate A. Estimate the percentage of supporters in the wider population, and comment on how confident this estimate is. | 50% — similar |
| 13 | A researcher wants to learn about Year 8 lunch preferences. Suggest a sampling plan that avoids bias. | Same logic |
| 14 | Identify whether each scenario is a census, a sample or biased: (a) UK government population census every 10 years. (b) Surveying every 10th student exiting the school. (c) Asking the football team about favourite sport. | Same. |
| 15 | A class of 30 students is surveyed about study time. The mean is 7.4 hours/week. Is this representative of all Y8 students? Suggest two reasons it might not be. | Same. |
| 16 | Improve this question: "How often do you eat the disgusting cafeteria food?" Provide a fairer version. | "What is your view on phones in school? Options: Allow / Restrict / Ban." |
| 17 | A sample of 40 students has 18 left-handed. The expected proportion is 10%. Comment on whether the sample is consistent with this. | 8% is close to 10% — consistent |
| 18 | A teacher conducts a survey on a school trip. List two reasons why this sample is unlikely to represent the whole school. | Same. |
| 19 | Identify the most representative sampling method for a school survey: (a) volunteers in the library; (b) random class draw; (c) every 10th student exiting; (d) stratified across year-groups. | Same. |
| 20 | A data set has the values 152, 148, 1500, 155 (heights in cm). Clean it and explain. | 220°C impossible for room temperature; correct to 22. |
| 21 | A school survey on phone use has 800 students. A stratified random sample of 80 is taken: Year 7 (200), Year 8 (180), Year 9 (220), Year 10 (100), Year 11 (100). How many from each year? | Apply 1/10 to each. |
| 22 | A sample of 30 students reports an average of 6.5 h homework per week with range 4. Estimate the school-wide average (school of 600 students) and discuss the confidence in this estimate. | Estimate 7.0 h; better confidence with n = 50 |
| 23 | For a survey about lunch preferences in a school of 1000 students, you want results accurate to ±3%. Approximately how large a sample is needed? | $n \approx 400$ |
| 24 | A school surveys 120 students about whether they bring their own lunch. 78 say yes. (a) What percentage in the sample? (b) Estimate the count in a year-group of 600. (c) Comment on accuracy. | (a) 65% (b) 520 (c) Margin ≈ 7% |
| 25 | A survey question is "Do you prefer cats over dogs?" State two ways this question is poorly designed and improve it. | Same. |
| 26 | A teacher considers two sampling plans for a school of 600. (i) 60 random students from the whole school; (ii) the first 60 students arriving at school. State which is better and why. | Same. |
| 27 | A doctor wants to estimate the average daily steps for the local town (~50 000 people). Suggest two methods, with pros and cons. | Same. |
| 28 | A school survey contains the question: "How many hours of TV do you watch a week? (a) 0–5, (b) 6–10, (c) 10–15, (d) more than 15." Identify the problem with the response options. | Same. |
| 29 | Sample with replacement vs without replacement. A class of 30 students has names placed in a hat. Five draws are made (with replacement). On average, how many distinct names are drawn? | Same. |
| 30 | In an opinion poll, respondents are asked: "How satisfied are you with the school cafeteria? Very satisfied / Satisfied / Neutral / Dissatisfied / Very dissatisfied." Identify whether this is biased. | Same. |
| 31 | A school of 800 students will be surveyed about transport. Year 7: 200, Year 8: 200, Year 9: 200, Year 10: 100, Year 11: 100. A stratified sample of 80 is required, but you want to ensure ≥ 5 from Year 11. How many from each year? | Same. |
| 32 | A sample of 50 dogs at a vet has mean weight 18 kg, range 30 kg. The vet treats 1000 dogs/year. Estimate the population mean and discuss the spread. | Same. |
| 33 | A polling firm surveys 1000 voters and reports candidate A at 51% support. Margin of error ±3%. What does this mean for confidence? | Same. |
| 34 | A pet-owner survey has 60% response rate (out of 500 sent). 200 responded "yes, owns a pet". Estimate the percentage of pet owners in the surveyed population. | Same. |
| 35 | A student conducts a survey at the school gate asking about phone use. Identify three types of bias possible and how to reduce each. | Same. |
| 36 | A stratified sample of 100 from a population of 1000 is split: 600 women and 400 men. After sampling, the survey finds 60 women say "yes" and 30 men. Estimate the overall population proportion of "yes" responses. | Same. |
| 37 | A class of 28 students completes a maths test. The teacher's class mean is 16/20 with range 7. The teacher reports "the school average is 16/20". Critique this claim. | Same. |
| 38 | A survey of 200 students yields a 95% confidence interval of "30–40% prefer pizza". Interpret this interval correctly. What does **not** it mean? | Same. |
| 39 | Design an unbiased question to estimate the proportion of students in a school who exercise daily, and state the sampling method. | Same. |
| 40 | A teacher cleans a dataset of 200 marks. She notices 15 zeros, 1 value of 100, and 5 missing entries. Outline a cleaning strategy. | Same. |
Problems — Worked Solutions
1
**Stratified sampling.** A school has 200 pupils in Year 7, 180 in Year 8, and 220 in Year 9. A survey of 90 pupils is to be taken using stratified random sampling. (a) How many from each year? (b) Compare with simple random sampling of 90 from the whole school. Which is more likely to over- or under-represent a year? (c) Suppose 20 new Year 7 pupils are added and the sample size stays 90. How does the Year-7 sample size change?
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
(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 $90/620 \approx 0.145$. Year 7 (now 220): $\approx 32$ pupils.
2
**Sampling bias scenarios.** For each scenario, state whether the sample is likely biased and explain. (a) Surveying mass transport use only by interviewing people on a tram. (b) Estimating average height from a basketball team. (c) Phone polling people about TV viewing habits at 9 am on a weekday.
Answer
(a) Biased (b) Biased (c) Biased
(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.
3
**Predicting from a sample.** A sample of 25 Year 8 pupils contains 3 left-handers. The Y8 cohort has 400 pupils. (a) Estimate the number of left-handed Y8 pupils. (b) Identify one assumption being made. (c) The school's overall stat is "about 10% of pupils are left-handed". Does your estimate agree? Discuss.
Answer
(a) 48 (b) Sample representativeness (c) Estimate (12%) is close to 10% — within sampling error
(a) Proportion 3/25 = 0.12. Estimate $0.12 \times 400 = 48$.
(b) The sample is assumed to be representative of the year-group (random, unbiased).
(c) 12% vs 10% — close. With $n = 25$, sampling error is ±5–10%, so consistent.
4
**Designing a survey.** You want to know how many Y8 pupils enjoy maths. Design a survey: (a) Write three good survey questions. (b) Specify a sample size and sampling method. (c) State two ways to reduce bias.
Answer
(a) Sample given in working (b) Stratified random, n = 60 (c) Anonymous; balanced response options
(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.
5
**Sample size and confidence.** Two polls predict election support: - Poll A: 60% support, sample size 100, MoE ±10%. - Poll B: 60% support, sample size 1000, MoE ±3%. (a) Which poll is more reliable? Why? (b) For Poll A, the true support could be anywhere from ? to ? (c) For Poll B, ditto. (d) The candidate needs > 50% to win. Which poll is more conclusive?
Answer
(a) Poll B (larger sample → smaller MoE) (b) 50–70% (c) 57–63% (d) Poll B (CI doesn't cross 50%)
(a) Larger sample reduces sampling error. MoE ∝ $1/\sqrt{n}$.
(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.
6
**Stratified vs simple random.** A school has the following enrolment: | Year | 7 | 8 | 9 | 10 | 11 | |------|---|---|---|----|----| | Count | 150 | 140 | 130 | 120 | 100 | A sample of 64 is required, stratified by year. (a) Compute the sample size for each year (round to nearest integer). (b) The school decides to oversample Y11 to ensure ≥ 15 are in the sample. Revise.
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
(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.
7
**Bias detection.** Identify the type of bias in each scenario and suggest a fix. (a) Asking "Don't you agree that the maths curriculum is great?" (b) Calling phone numbers only during work hours. (c) Surveying only students at the after-school club about extracurriculars.
Answer
(a) Leading wording (b) Time-of-day bias (c) Selection bias
(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.
8
**Cleaning data.** A spreadsheet of student heights (cm) includes: 152, 148, 1500, NA, 153, 155, "?", 149, 151, 1.6. (a) Identify problematic entries. (b) Suggest a cleaning plan. (c) Compute the mean before and after cleaning.
Answer
(a) 1500, NA, "?", 1.6 are problematic (b) Correct or remove (c) Before: undefined or ≈ 207; after ≈ 151
(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 = $1218/8 = 152.25$ cm.
9
**Sample-size planning.** A school of 800 students wants to estimate the percentage who walk to school, to within ±5%. (a) Estimate the required sample size (rule of thumb: $n \approx 1/p^2$ where $p$ is the desired margin in decimal form). (b) Compare with sampling 30 students. What is the approximate margin of error?
Answer
(a) ≈ 400 (b) ±18% — too imprecise
(a) Margin of error $\approx 1/\sqrt{n}$. Set $= 0.05$: $n = 400$.
(b) For $n = 30$: $1/\sqrt{30} \approx 0.18$ → MoE ≈ ±18%. Far too imprecise to make any meaningful conclusion.
10
**Non-response bias.** A survey is sent to 500 students, but only 200 respond. (a) Compute the response rate. (b) Why is non-response a problem for estimates? (c) Suggest two ways to reduce non-response.
Answer
(a) 40% (b) Non-respondents may differ from respondents (c) Reminders, incentives
(a) $200/500 = 40\%$.
(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.
11
**Polling election support.** A poll surveys 1000 voters: 540 support A, 460 support B. The election needs >50% to win. (a) Find the sample percentages. (b) State the margin of error (rule of thumb). (c) Can the poll predict the winner with confidence?
Answer
(a) A 54%, B 46% (b) MoE ≈ ±3% (c) Yes — A's 54% is outside the 50% threshold + MoE
(a) 54% A, 46% B.
(b) $1/\sqrt{1000} \approx 0.032$ → MoE ±3.2%.
(c) A: 54% ± 3.2% → 50.8%–57.2%. Entirely above 50% → A predicted winner with confidence.
12
**Question design audit.** A school survey asks: "How often do you study at home?" (a) Suggest 5 mutually exclusive, exhaustive response options. (b) Critique the question — what assumptions does it make? (c) Rewrite the question to remove the assumption.
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)"
(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.