AQA A-Level Mathematics: K: Statistical sampling (7357) – Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – examination boards hold copyright in their own papers. Use these alongside the official past papers from your board or school.

These questions cover Section K: Statistical sampling (K1) of the AQA A-level Mathematics (7357) specification, version 1.3, for A-level exams from June 2018 onwards. Section K is Paper 3 content in the specification. A calculator is required in every 7357 paper, so every question here assumes you have one. Questions 9 to 11 are extended problems that combine several parts of K1.

Learn the methods first in the Statistical sampling study guide and the Statistical sampling revision notes. The 7357 course hub and the printable checklist show where this topic fits, and the free 10-minute diagnostics help you find weak spots.

Questions

1. A cinema chain wants to know how satisfied its 15,000 loyalty-card holders are. It emails a questionnaire to 300 of them. State the population and the sample. [2]

2. A factory makes fireworks. Give two reasons why it tests a sample of each batch rather than carrying out a census. [2]

3. A raffle has 840 tickets, numbered 1 to 840. Describe how to use a random number generator to choose a simple random sample of 50 tickets. [3]

4. To find out how much time adults in a city spend reading, a student asks the first 40 adults who leave the central library on a Saturday afternoon.

(a) Name this sampling method. [1] (b) Explain why the results are unlikely to be representative, stating the likely effect on the estimate. [2]

5. A warehouse has 960 parcels on an ordered dispatch list. A systematic sample of 40 parcels is to be checked.

(a) Find the sampling interval. [1] (b) The random start is 15. List the first three parcel numbers chosen and the last one. [2] (c) Explain why this is not a simple random sample. [1]

6. A tennis club has 84 junior, 196 adult and 70 senior members.

(a) Find how many of each group should be in a stratified sample of 25. [3] (b) State one advantage of stratified sampling over simple random sampling here. [1]

7. A random sample of 40 bags of flour is taken from a batch of 2,000 bags. 6 of the sampled bags have a mass below 500 g. The total mass of the 40 sampled bags is 20,180 g.

(a) Estimate the number of bags in the batch with a mass below 500 g. [2] (b) Estimate the mean mass of a bag in the batch. [1] (c) The supplier claims fewer than 1 in 10 bags in a batch are below 500 g. Comment on this claim using the sample. [2]

8. Two students each take a random sample of 12 pupils from the same sixth form and record the hours of paid work done last week. Ana’s sample mean is 3.2 hours. Ben’s is 3.9 hours.

(a) Explain why the two sample means are different. [2] (b) Suggest two ways the students could get a more reliable estimate of the population mean. [2]

9. A town council wants the views of the town’s 24,000 adult residents on a new bus route. Three methods are suggested:

• P: interview adults waiting at the bus station on one weekday morning;
• Q: choose a simple random sample of adults from the electoral register and post them a questionnaire;
• R: split the town into its six wards and take a stratified sample by ward.

(a) Give two criticisms of method P. [3] (b) Give one advantage and two disadvantages of method Q. [3] (c) Explain why the council might prefer R to Q. [2]

10. A factory runs three shifts. The morning shift makes 240 items, the afternoon shift 180 items and the night shift 120 items per day. A quality manager wants a stratified sample of 45 items from one day’s output.

(a) Find how many items to take from each shift. [2] (b) Describe how the items from the morning shift should be chosen. [2] (c) The numbers of defective items found are 3 (morning), 2 (afternoon) and 4 (night). Show that the estimated proportion of defective items in the day’s output is 20%. [3] (d) Comment on the night shift’s results. [1] (e) A colleague suggests a quota sample instead, taking the first items available from each shift. Explain why this could give a biased estimate. [2]

11. A researcher wants the views of Year 12 pupils in a region with 40 secondary schools.

(a) Describe how a cluster sample could be taken. [2] (b) Give one advantage of cluster sampling here and one reason the estimates may be less reliable than from a stratified sample by school. [2] (c) State the sampling frame needed for a simple random sample of Year 12 pupils in the region. [1]

Answers

1. Population: all 15,000 loyalty-card holders [1]. Sample: the 300 card holders sent the questionnaire [1]. [2] Examiner insight: the population must be the exact group in the question; “cinema customers” loses the mark because it is wider than the card holders.

2. Testing destroys the firework, so a census would leave nothing to sell [1]. A sample is quicker and cheaper to test [1]. [2] Examiner insight: each reason must be a separate point tied to the context; two versions of “cheaper” earn one mark only.

3. Use the numbers 1 to 840 as the sampling frame (the ticket numbers) [1]. Generate random integers from 1 to 840 with a calculator or computer [1]. Select the matching tickets, ignoring repeats, until 50 are chosen [1]. [3] Examiner insight: “pick 50 at random” scores nothing; the marks are for the frame, the random number generation, and dealing with repeats.

4. (a) Opportunity sampling [1]. (b) Library users are more likely to read than other adults [1], so the time spent reading will be overestimated [1]. Examiner insight: a critique needs the group over-represented and the direction of the effect; “it is biased” alone gets no credit.

5. (a) k = 960 ÷ 40 = 24 [1]. (b) First three: 15, 39, 63 [1]. Last: 15 + 39 × 24 = 951 [1]. (c) Once the start is chosen, the other parcels are fixed, so not every possible sample of 40 can be chosen [1]. Examiner insight: in (b) using 40 × 24 instead of 39 × 24 for the last item is a common slip; the 40th item is r + 39k.

6. (a) Total = 84 + 196 + 70 = 350 [1]. Juniors 84/350 × 25 = 6, adults 196/350 × 25 = 14 [1], seniors 70/350 × 25 = 5 [1]. (b) Each age group is represented in proportion to its size [1]. Examiner insight: show the fraction for each stratum; three bare whole numbers with no working risk losing the method mark.

7. (a) Sample proportion 6/40 = 0.15 [1]. Estimate 0.15 × 2000 = 300 bags [1]. (b) 20180 ÷ 40 = 504.5 g [1]. (c) The sample proportion 0.15 is more than 0.1, so the sample does not support the claim [1]. But it is only one sample of 40, and another sample could give a different proportion, so this is not conclusive [1]. Examiner insight: in (c) one mark is for comparing with the claim and the other for a sensible caveat about sampling variability; stating “the claim is false” overstates the evidence.

8. (a) The samples contain different pupils [1], and with small samples chance differences between samples are large (sampling variability) [1]. (b) Use a larger sample [1]. Combine the two samples into one of 24 [1]. Examiner insight: “they made a mistake” earns nothing; credit comes from the idea that different random samples give different results, linked to the small sample size.

9. (a) Only bus users are asked, so support for a new bus route is likely to be overestimated [1]. Residents who drive, walk or cycle are under-represented [1]. One weekday morning misses people who travel at other times, such as shift workers [1]. (b) Advantage: every adult on the register has an equal chance of selection, so there is no selection bias [1]. Disadvantages: postal questionnaires may get a low response rate [1]; those who reply may hold stronger views than those who do not [1]. (c) Views may differ between wards, for example near and far from the proposed route [1]. Stratifying by ward makes sure each ward is represented in proportion to its adult population [1]. Examiner insight: in (a) three marks need three distinct, context-linked points; repeating “not representative” in different words earns one mark at most.

10. (a) Total 540. Morning 240/540 × 45 = 20, afternoon 15 [1], night 10 [1]. (b) Number the 240 morning items [1]. Use random numbers to select 20, ignoring repeats [1]. (c) Estimated defective: morning 3/20 × 240 = 36, afternoon 2/15 × 180 = 24, night 4/10 × 120 = 48 [1]. Total 36 + 24 + 48 = 108 [1]. 108/540 = 0.2 = 20% [1]. (d) The night shift’s sample defect rate (40%) is much higher than the others (15% and about 13%), which suggests a problem on that shift worth investigating, though it is based on only 10 items [1]. (e) The first items available are not chosen at random [1], so if, for example, the first items of a shift are made while machines are warming up, they may have a different defect rate from the rest [1]. Examiner insight: (c) is a “show that”, so each step must appear; quoting 9/45 = 0.2 without explaining why it matches the weighted estimate risks losing marks.

11. (a) Choose some of the 40 schools at random [1]. Survey all (or a random sample of) Year 12 pupils in the chosen schools [1]. (b) Advantage: it is cheaper because researchers visit only a few schools [1]. Less reliable: schools may differ from each other, and unchosen schools contribute nobody [1]. (c) A list of all Year 12 pupils in the region’s 40 schools [1]. Examiner insight: in (a) the random choice of clusters must be stated; choosing “nearby” schools turns it into an opportunity sample and loses the mark.

Where marks are usually lost

• Describing the population too broadly or too narrowly for the context.
• Writing “choose at random” without a sampling frame, numbering or a random number generator.
• Leaving out “ignore repeats” when selecting by random numbers.
• Getting the last systematic item wrong by using n × k instead of (n − 1) × k.
• Stratum sizes that do not add up to the sample size after rounding.
• Calling a criticism “biased” without saying who is over-represented and which way the estimate moves.
• Treating a sample estimate as proof about the population.
• Describing quota sampling as random within groups.
• Giving two criticisms that are really the same point.

Next steps

• Recap with the Statistical sampling revision notes.
• Re-read the Statistical sampling study guide for any method you missed.
• See every topic on the 7357 course hub.
• Tick off outcomes on the printable checklist.
• Try one of the free 10-minute diagnostics.
• Book a free trial class.

Official syllabus

AQA A-level Mathematics (7357) specification, version 1.3 (31 January 2018), for A-level exams June 2018 onwards, published by AQA. Section 3.12, K: Statistical sampling.