S1 Statistical sampling — revision question pack

1 specification points · notes, questions, answers and worked methods

Checked against Edexcel 9MA0 section S1. Review basis: the qualification registry sourced from the Pearson Edexcel Level 3 Advanced GCE in Mathematics (9MA0) specification; registry verification recorded 11 July 2026.

How this checking works

S1.1 · Understand and use the terms 'population' and 'sample'; use samples to make informal inferences about the population; use sampling techniques including simple random and opportunity sampling; select or critique techniques in context.

Explanation

The population is the complete set about which an inference is required; a sample is the subset from which data are actually collected.
A simple random sample gives every member an equal selection chance; systematic sampling uses a fixed interval after a random start, while stratified sampling preserves chosen population proportions.
Quota sampling fills category targets without random selection, whereas opportunity sampling uses whoever is available; both are quick but vulnerable to selection bias.
When drawing an informal inference, discuss representativeness, sampling frame, non-response and sample size; a large biased sample is not automatically reliable.
A census avoids sampling variation but may be slow, costly or impractical; a sample is faster and cheaper but may be unrepresentative, and different valid samples can lead to different conclusions.

Worked example

A theatre has a numbered list of its $1260$ members. Describe how to select a simple random sample of $60$ members.

1.Use the complete membership list as the sampling frame.
2.Assign each member one unique number, generate numbers uniformly from the full range, and ignore repeats until $60$ different labels have been obtained.
3.This gives each member an equal chance of selection.

Answer: Label the members $1$ to $1260$ .; Use a random-number generator to obtain $60$ distinct integers from $1$ to $1260$ .; Select the members with those labels.

Common mistakes

Don't use an opportunity sample from one class and generalise its result to the whole school.
Don't select numbers with replacement or from the wrong range, so some members are duplicated or impossible.

Exam tip

For a simple random sample, specify the numbered sampling frame, a random generator and rejection of repeats or out-of-range values.

Tier 1 · Easy

1.
A college wants to estimate the weekly study time of all $1840$ students. It records the study time of $75$ students. State the population and the sample.
(2)
(Total for Question 1 is 2 marks)
2.
A conservation officer records the condition of every one of the $86$ trees in a park. State whether the officer has used a sample or a census, and give one advantage of this choice.
(2)
(Total for Question 2 is 2 marks)

Tier 2 · Standard

1.
A student surveys the first $40$ people leaving the school cafeteria at lunchtime about the quality of all school meals. Identify the sampling method, give two reasons the sample may be biased, and suggest one improvement.
(4)
(Total for Question 1 is 4 marks)
2.
A gallery has $720$ adult members and $280$ junior members. It takes a proportional stratified sample of $50$ members. Find how many members should be selected from each group and describe how the individuals could be chosen.
(3)
(Total for Question 2 is 3 marks)
3.
A warehouse has an ordered sampling frame of $960$ employees. Describe how to take a systematic sample of $80$ employees, including how the first employee is selected and the labels of the next two employees if the random start is $7$ .
(4)
(Total for Question 3 is 4 marks)

Tier 3 · Hard

1.
To estimate support for extending library opening hours, a researcher asks the first $120$ people entering the library after $8$ pm. Of these, $96$ support the proposal. Critique the sampling method and the inference that about $80\%$ of all town residents support the proposal.
(5)
(Total for Question 1 is 5 marks)
2.
A rail operator places a survey link in its delay-alert app. Of the $300$ passengers who choose to respond, $219$ support adding late-evening services. A director concludes that about $73\%$ of all passengers support the change. Evaluate this conclusion and suggest a better sampling procedure.
(5)
(Total for Question 2 is 5 marks)
3.
A sixth form contains $270$ Year 12 science students, $198$ other Year 12 students, $196$ Year 13 science students and $136$ other Year 13 students. A proportional stratified sample of $75$ students is required using these four strata. Calculate suitable integer allocations, showing how you keep the total at $75$ . State how the students should be selected within each stratum and give one practical difficulty with carrying out the plan.
(5)
(Total for Question 3 is 5 marks)
4.
A town has $2400$ registered sports-centre users: $960$ are aged under $25$ , $720$ are aged $25$ to $44$ , $480$ are aged $45$ to $64$ and $240$ are aged $65$ or over. An interviewer must obtain a proportional quota sample of $100$ users. Calculate the four quotas. The interviewer plans to approach convenient users until every quota is filled. Explain why meeting the quotas does not make this a random sample, and describe how a proportional stratified random sample would differ.
(6)
(Total for Question 4 is 6 marks)
5.
A production line fills $1200$ vials in a repeating order from six filling heads, labelled $A,B,C,D,E,F,A,B,\ldots$ . A technician proposes a systematic sample of $60$ vials, using a random start and then every $20$ th vial. Find the sampling interval and explain why this plan may fail to represent all six heads. Propose a sampling procedure that gives every vial an equal chance of selection and state one practical requirement for carrying it out.
(5)
(Total for Question 5 is 5 marks)

Answer key

Answers begin on a new printed page so the question pack can be completed without the solutions alongside it.

S1.1 · Understand and use the terms 'population' and 'sample'; use samples to make informal inferences about the population; use sampling techniques including simple random and opportunity sampling; select or critique techniques in context.

Tier 1 · Easy

Mark scheme for S1.1 Tier 1 · Easy
Question	Scheme	Marks
1	Population: all $1840$ students at the college. Sample: the $75$ students whose study times were recorded.	2
(2 marks)		2
Notes
Identify the whole group about which the estimate is required: all $1840$ students. The observed subset consists of the $75$ students, so that subset is the sample.
2	A census There is no sampling variation because every tree in the population is included, so no part of the park is omitted from the recorded population.	2
(2 marks)		2
Notes
The population is all $86$ trees in the park. Since every member of that population is inspected, the process is a census rather than a sample and avoids variation caused by selecting only a subset.

Tier 2 · Standard

Mark scheme for S1.1 Tier 2 · Standard
Question	Scheme	Marks
1	Opportunity sampling It excludes pupils who do not use the cafeteria, such as pupils who bring lunch. It may over-represent particular year groups or pupils whose lunch ends at that time. Use a sampling frame of all pupils and select a random sample, or sample across different days and times.	4
(4 marks)		4
Notes
The interviewer uses whoever is conveniently available first, so this is an opportunity sample. Cafeteria users may have systematically different views from pupils who bring food, and the chosen time may over-represent particular year groups or timetables. A simple random sample from the whole-school roll would give each pupil an equal chance; sampling across several times is a weaker but practical improvement.
2	$36$ adult members and $14$ junior members Number the members within each group and use a random-number generator to select the required number of distinct labels from each group.	3
(3 marks)		3
Notes
There are $1000$ members. The adult allocation is $50(720/1000)=36$ and the junior allocation is $50(280/1000)=14$ . Random selection within each stratum avoids choosing only the most readily available members.
3	Sampling interval $=960/80=12$ Choose the first label at random from $1$ to $12$ . For random start $7$ , select labels $7,19,31,\ldots,955$ .	4
(4 marks)		4
Notes
Divide the population size by the required sample size to obtain the interval $12$ . A random start from the first interval prevents the starting position from being chosen subjectively. Repeatedly add $12$ : after $7$ , the next two labels are $19$ and $31$ , and the eightieth is $7+79(12)=955$ .

Tier 3 · Hard

Mark scheme for S1.1 Tier 3 · Hard
Question	Scheme	Marks
1	This is an opportunity sample. Late-evening library users are likely to be more supportive than residents who do not use the library then, so the sample is biased. The sampling frame excludes many town residents, and repeat/non-response details are unknown. Therefore the sample proportion $96/120=0.80$ should not be generalised to all town residents.	5
(5 marks)		5
Notes
The participants were chosen because they were available at one place and time, so the method is opportunity sampling. The selection mechanism over-represents people who already use the library late. Although the observed sample proportion is $96/120=0.80$ , its bias matters more than its size, so it does not justify the stated population inference. A simple random sample from a suitable register of residents would be more representative.
2	The response proportion is $219/300=0.73$ . The respondents form a voluntary-response sample, an extreme form of opportunity sampling, so passengers with strong opinions may be more likely to reply. Passengers who do not use the app and app users who do not respond may have systematically different views, so the response proportion need not represent all passengers. Take a stratified random sample of passenger journeys across routes and time periods, contact the selected passengers directly and follow up non-respondents.	5
(5 marks)		5
Notes
Although $219/300=0.73$ , this is a voluntary-response sample, which is an extreme form of opportunity sampling rather than a random sample. Self-selection, undercoverage of passengers who do not use the app and non-response can all bias the result. Random selection across routes and time periods, followed by direct contact with the selected passengers, provides a more defensible basis for inference.
3	The unrounded allocations are $25.3125$ , $18.5625$ , $18.375$ and $12.75$ , respectively. Suitable integer allocations are $25$ , $19$ , $18$ and $13$ , which sum to $75$ . After allocating the integer parts, the two remaining places go to the strata with the largest fractional remainders: $0.75$ and $0.5625$ . Use an accurate register carrying both stratum labels, then select the required number of distinct students at random within each stratum. A practical difficulty is that subject or year-group records may be missing or out of date, so students could be placed in the wrong stratum; non-response may also disturb the planned proportions.	5
(5 marks)		5
Notes
The population size is $270+198+196+136=800$ . Multiplying each stratum size by $75/800$ gives $25.3125$ , $18.5625$ , $18.375$ and $12.75$ . Their integer parts total $73$ , so assign the two remaining places to the largest remainders, giving $25,19,18,13$ . Random selection within each labelled stratum avoids interviewer choice, but the procedure relies on a current sampling frame and still needs a plan for non-response.
4	The quotas are $40$ , $30$ , $20$ and $10$ , respectively. Filling quotas by approaching convenient users is quota sampling; selection within each age category is non-random and may favour users attending at particular times or places. Matching the age proportions removes neither selection bias within the categories nor possible non-response bias. For a proportional stratified random sample, use a sampling frame for each age category and select $40$ , $30$ , $20$ and $10$ distinct users at random from the respective frames.	6
(6 marks)		6
Notes
Multiply $100$ by each population proportion: $100(960/2400)=40$ , $100(720/2400)=30$ , $100(480/2400)=20$ and $100(240/2400)=10$ . Quota sampling fixes category totals but permits interviewer choice within a category. Stratified random sampling uses the same allocations but replaces convenience selection with random selection from complete category frames.
5	The sampling interval is $1200/60=20$ . Since adding $20$ changes the head position by $20\equiv2\pmod 6$ , the sample cycles through only three of the six filling heads. A fault affecting an omitted head could therefore be missed, so the periodic production order makes this systematic plan unrepresentative. Number all $1200$ vials and use a random-number generator to select $60$ distinct labels, rejecting repeats; this requires a complete, accurate sampling frame linking labels to vials.	5
(5 marks)		5
Notes
Systematic sampling would normally use interval $k=N/n=20$ . Here the population order has period $6$ , and repeated addition of $20$ is repeated addition of $2$ modulo $6$ , so only three residue classes occur before the sequence repeats. A simple random sample from all $1200$ labels avoids locking the sample to this production cycle, provided the full frame is available and current.