These are original Marlbridge practice questions, not past exam questions. All places, organisations and data are fictional. The rubrics are a suggested Marlbridge rubric to help you check your work; they are not an official scoring guideline. Conventions: “direction” means whether the statistic is likely to be too high or too low compared with the parameter; round percentages to 1 decimal place.
Question 1 (multiple choice · foundation)
A fitness magazine wants to know what proportion of its readers exercise at least four times a week. It prints a link to an online survey in one issue and invites readers to take part. In total, 2,300 readers complete the survey. Which kind of bias is the main concern?
- (A) Nonresponse bias
- (B) Undercoverage bias
- (C) Voluntary response bias
- (D) Response bias caused by question wording
Answer and explanation
Answer: (C). The readers chose themselves. A sample made up entirely of volunteers gives voluntary response bias; keen exercisers are more likely to take part, so the proportion is likely too high.
- (A) Nonresponse needs a sample the researcher chose, some of whom did not reply. Here readers volunteered.
- (B) Readers without internet access may be missed, but the main problem is that the whole sample is self-selected.
- (D) Nothing is said about how the question was worded, so there is no evidence of wording bias.
Question 2 (multiple choice · core)
The council of the fictional city of Draymoor wants to estimate the proportion of its adult residents who drive to work. It chooses a simple random sample of 600 people from the list of residents who own a registered car and interviews all of them. Which statement best describes the sampling method?
- (A) Undercoverage bias; the estimate is likely to be too high.
- (B) Undercoverage bias; the estimate is likely to be too low.
- (C) Voluntary response bias; the estimate is likely to be too high.
- (D) No bias, because the sample is a simple random sample.
Answer and explanation
Answer: (A). Residents without a car are not on the list, so they cannot be chosen: undercoverage. They rarely drive to work, so leaving them out makes the sample proportion too large.
- (B) has the right bias but the wrong direction: the missing group drives less.
- (C) Nobody volunteered: the council chose the sample at random from a list.
- (D) Selection is random only among people on the list; an incomplete list still gives undercoverage.
Question 3 (multiple choice · core)
A fictional streaming service, Flicker, wants to know what proportion of its subscribers would pay more for a plan with no adverts. It considers two plans:
- Plan 1: a pop-up poll on its home page that any visitor can answer. Last time, a poll like this got 12,000 replies.
- Plan 2: a simple random sample of 400 subscribers from its full subscriber list, each sent a short survey with a reminder.
Which statement is correct?
- (A) Plan 1 is better, because 12,000 replies is far more than 400.
- (B) Plan 1 is likely to be biased whatever its size, while Plan 2 avoids the voluntary response bias of the poll.
- (C) Both plans are equally good, because both samples are large.
- (D) Plan 1 cannot be biased, because every visitor has the chance to answer.
Answer and explanation
Answer: (B). Plan 1 is a voluntary response sample: visitors decide whether to answer, so people with strong views are over-represented. A bigger self-selected sample is still biased. Plan 2 uses chance to choose subscribers from a complete list, so selection is not biased (some nonresponse is still possible, which is why the reminder helps).
- (A) assumes a larger sample removes bias. Size reduces chance variation, not bias.
- (C) ignores how the samples were chosen.
- (D) confuses “anyone may answer” with “chance decides who is chosen”. In a voluntary poll, people decide for themselves.
Question 4 (constructed response · core)
At the fictional Westerly Sixth Form College, 400 students were split at random into two groups of 200. Each group was asked one version of a question about the lunch break.
- Version A: “Given how much a longer lunch break would improve students’ well-being, do you support a longer lunch break?” 72% answered yes.
- Version B: “Do you support or oppose a longer lunch break?” 49% said they support it.
(a) How many students in each group supported a longer lunch break? (b) Identify the kind of bias shown, say which version causes it, and give its direction. (c) The college wants to estimate the proportion of all its students who support a longer lunch break. Which result should it use? Explain.
Worked solution and suggested Marlbridge rubric
(a) Version A: 0.72 × 200 = 144 students. Version B: 0.49 × 200 = 98 students.
(b) Response bias from question wording. Version A is leading: it says a longer break would “improve students’ well-being” before asking, pushing students towards “yes”. So Version A’s 72% is likely to be too high. The groups were formed at random, so the 23-point gap is far larger than chance alone would explain; the wording is the obvious cause.
(c) Version B (49%). It names both options (“support or oppose”) and gives no reason to favour either, so it is the neutral wording.
| Point | What earns it |
|---|---|
| 1 | Correct counts, 144 and 98 |
| 1 | Names response bias (question wording) and identifies Version A as leading, quoting or describing the leading phrase (“Version A is biased” alone does not earn this) |
| 1 | States that Version A’s result is likely too high, linked to the wording |
| 1 | Chooses Version B with a reason about neutral wording |
Question 5 (constructed response · core)
The fictional Riverside Sports Centre wants to estimate the mean number of visits per week made by its members. It emails a questionnaire to a simple random sample of 250 members chosen from its full membership list. The questionnaire asks, “How many times a week do you usually visit the centre?” Ninety-five members reply. Their mean answer is 3.2 visits per week.
(a) Calculate the response rate. (b) Identify two potential sources of bias. For each, explain how it arises in this context and whether it is likely to make 3.2 too high or too low. (c) The centre’s entry-gate records show that the mean for all members over the same period was 1.9 visits per week. Is this consistent with your answer to (b)? Explain briefly.
Worked solution and suggested Marlbridge rubric
(a) 95 ÷ 250 = 0.38, so the response rate is 38%. That means 155 chosen members did not reply.
(b)
- Nonresponse bias. Frequent visitors are more involved with the centre and more likely to answer its email; rare visitors are more likely to ignore it. So 3.2 is likely too high.
- Response bias from self-reported answers. Members report their “usual” visits from memory. People tend to overstate healthy habits and remember their best weeks, so 3.2 is again likely too high.
(c) Yes. Both sources point upwards, and 3.2 is 1.3 visits per week above 1.9. One sample cannot show which source caused the gap, but it is in the predicted direction.
| Point | What earns it |
|---|---|
| 1 | Response rate 38% |
| 1 | Nonresponse bias explained in context (who is less likely to reply) |
| 1 | A second, different source (self-report response bias, or another valid source) explained in context |
| 1 | A justified direction (too high) for both sources |
Selection is not a source of bias here: it was a simple random sample from the full list. Point 4 needs a reason, not just “too high”.
Question 6 (constructed response · stretch)
The council of the fictional town of Kelsbury wants to estimate the proportion of its households that support a new charge for parking in the town centre. Two plans are proposed.
- Plan A: a poll on the council website, open for two weeks. A similar poll last year received 1,900 responses.
- Plan B: a simple random sample of 300 addresses from the council’s complete address list. An interviewer visits each address up to three times and asks one adult a neutral question. A pilot reached 246 of 300 households (82%).
(a) Identify the main source of bias in Plan A and explain the likely direction of the bias. (b) A councillor says, “Plan A is better because 1,900 is much more than 300.” Respond to this claim. (c) Identify one potential source of bias that remains in Plan B and explain how it could arise. (d) Which plan should the council use? Justify briefly.
Model answer and suggested Marlbridge rubric
(a) Voluntary response bias. People choose whether to answer the poll. Those who feel strongly, especially those angry about a new charge, are the most likely to take the trouble. So the proportion who support the charge is likely to be too low.
(b) The claim is wrong. Plan A’s sample, however large, is made up of volunteers, so it is biased. More responses give a more precise estimate of the wrong group. Plan B’s smaller random sample avoids bias in who is chosen.
(c) Nonresponse bias. 18% of chosen households (54 of 300 in the pilot) could not be reached or refused. Households where adults work long hours are harder to find at home, and they may use town-centre parking differently, so their views on the charge may differ. Accept instead response bias: face to face with a council interviewer, some adults may give the answer they think the council wants.
(d) Plan B. It chooses households by chance from a complete list, follows up non-respondents and asks a neutral question.
| Point | What earns it |
|---|---|
| 1 | Names voluntary response bias in Plan A and gives a reasoned direction (support likely too low) |
| 1 | Explains that sample size does not remove bias |
| 1 | Identifies a valid remaining source in Plan B, explained in context |
| 1 | Chooses Plan B with a reason based on how the sample is chosen |
In (a), accept “too high” if it is argued in context (for example, town-centre residents who want fewer parked cars are more motivated to respond).
Question 7 (explanation · stretch)
At the fictional Ashdown School, records show that the mean number of days absent last term, over all 900 students, was 4.0 days. A statistics class tested two estimation methods, each used 8 times with a new simple random sample of 40 students. Every chosen student took part.
- Method P: each chosen student privately looks up their own record on the school portal and types the number into an anonymous online form.
- Method Q: each chosen student is asked by their form tutor, in front of the class, to say from memory how many days they missed.
The 8 sample means (days) were:
| Method | Sample means |
|---|---|
| P | 3.8, 4.3, 4.1, 3.6, 4.4, 3.9, 4.2, 3.7 |
| Q | 3.1, 3.4, 2.9, 3.3, 3.2, 3.0, 3.5, 3.2 |
(a) Which method appears to be biased? Give its direction and use the data as evidence. (b) Name the most likely source of this bias and explain how it arises. (c) A student says, “Method P is biased too, because none of its 8 results is exactly 4.0.” Respond. (d) Would using samples of 120 students with Method Q remove the bias? Explain.
Model answer and suggested Marlbridge rubric
(a) Method Q, which underestimates. All 8 of its sample means (2.9 to 3.5 days) are below the parameter of 4.0 days, and their mean is 3.2 days, about 0.8 days too low.
(b) Response bias. The answers are self-reported from memory, in front of a teacher and classmates. Students are likely to understate absences to avoid looking bad, and may forget some days. Both push answers down.
(c) The student is wrong. Method P’s results (3.6 to 4.4 days) fall on both sides of 4.0, four above and four below, and their mean is 4.0 days. That is chance variation, not bias, which means missing consistently in one direction.
(d) No. Larger samples would make Method Q’s results vary less, but students would still understate their absences. The results would cluster more tightly around about 3.2 days, still too low.
| Point | What earns it |
|---|---|
| 1 | Method Q, too low, with evidence from the data (all 8 below 4.0, or mean 3.2) |
| 1 | Response bias from self-reports or the public, face-to-face setting, explained in context |
| 1 | Explains that Method P shows chance variation on both sides of 4.0, not bias |
| 1 | Larger samples do not remove bias, with a reason |
How did you do?
- Q1 or Q3 wrong: re-read “Nonrandom samples: convenience and voluntary response” in the study guide.
- Q2 wrong: revisit “Undercoverage bias”.
- Q4 or Q7(b) wrong: revisit “Response bias” and Worked examples 1(c) and 2.
- Q5 wrong: revisit “Nonresponse bias” and Worked example 2.
- No directions given: re-read “Writing about bias”.
- Q3(A), Q6(b), Q7(c) or Q7(d) wrong: revisit “Bias: a fault in the method, not bad luck” and Figure 1.
Then tick off the topic checklist.