Statistical sampling – A-level Maths exam questions with mark scheme

1CoreCalculator4 marks

A company has 800 employees. The manager wants to find out how long employees spend commuting each day. She has an alphabetical list of all 800 employees and decides to take a systematic sample of 40 employees.

(a) State what is meant by a sampling frame. [1]

(b) Explain how the manager should take her systematic sample of 40 employees. [2]

(a) a list (or map) of all the sampling units / all members of the population (from which the sample is chosen)B1
(b) $800\div 40=20$, i.e. take every 20th employee from the (alphabetical) list (allow $k=20$)B1
(b) the first employee is chosen at random from the first 20 on the list (e.g. random number 1–20), then every 20th after thatB1
(c) any one of: quicker / cheaper / less time-consuming; easier to collect and process the data than a census of all 800 (accept: easier to manage). Ignore extra reasons.B1
Answer: (a) list of the whole population; (b) random start in 1–20, then every 20th; (c) e.g. quicker/cheaper
Examiner tip: In (b) the second mark needs the word random for the starting point – 'start at the first name' is not systematic sampling.

2CoreCalculator4 marks

A secondary school has 800 students. The numbers of students in each year group are shown below.

Year	7	8	9	10	11
Number of students	186	174	162	150	128

The head teacher wants a stratified sample of 60 students.

(a) Find the number of students that should be sampled from each year group. [2]

(b) State one advantage of using a stratified sample rather than a simple random sample in this situation. [1]

(a) uses $\frac{\text{year size}}{800}\times 60$ for at least one year, e.g. $\frac{186}{800}\times 60=13.95$M1
(a) 14, 13, 12, 11, 10 (must be whole numbers, total 60)A1
(b) each year group is represented in proportion to its size / sample reflects the structure of the population (allow: guarantees every year group is included in the sample; accept 'more representative' only if linked to the year groups)B1
(c) quota sampling is not random, so it may introduce (interviewer) bias / not every student has a chance of being selected (allow: interviewer chooses who is asked, so sample may not be representative; ignore 'harder/slower')B1
Answer: (a) Years 7–11: 14, 13, 12, 11, 10; (b) e.g. each year group represented in proportion to its size; (c) e.g. quota sampling is not random, so may be biased
Examiner tip: Round the decimals to whole students and check they still add up to 60 – leaving 13.95 students loses the A1.

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