What is S1.1: Sampling techniques?

The population is the complete set about which an inference is required; a sample is the subset from which data are actually collected.

How many marks are S1.1 questions worth?

S1.1 practice drills on this page are worth 2–6 marks each; marks in a real exam vary by question.

Which past exam papers has S1.1 appeared in?

We have not yet indexed a verified past-paper appearance for S1.1 — check the Edexcel A-level Maths 9MA0 past-papers hub directly.

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.

Draft — not yet indexed

Sampling techniques

Worked answers and methods for S1.1 on Edexcel A-level Maths 9MA0.

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.

Worked practice

Tier 1 · Easy

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)

Mark scheme

Mark scheme for question 1
Question	Scheme	Marks
1	Population: all $1840$ students at the college. Sample: the $75$ students whose study times were recorded.	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 marks)

Tier 2 · Standard

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 2 is 4 marks)

Mark scheme

Mark scheme for question 2
Question	Scheme	Marks
2	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
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.

(4 marks)

Tier 3 · Hard

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 3 is 5 marks)

Mark scheme

Mark scheme for question 3
Question	Scheme	Marks
3	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
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.

(5 marks)

Tier 1 · Easy

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 4 is 2 marks)

Mark scheme

Mark scheme for question 4
Question	Scheme	Marks
4	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
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.

(2 marks)

Tier 2 · Standard

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 5 is 3 marks)

Mark scheme

Mark scheme for question 5
Question	Scheme	Marks
5	$36$ 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
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 marks)

Tier 3 · Hard

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 6 is 5 marks)

Mark scheme

Mark scheme for question 6
Question	Scheme	Marks
6	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
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.

(5 marks)

Tier 2 · Standard

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 7 is 4 marks)

Mark scheme

Mark scheme for question 7
Question	Scheme	Marks
7	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
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 $7+79(12)=955$ .

(4 marks)

Tier 3 · Hard

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 8 is 5 marks)

Mark scheme

Mark scheme for question 8
Question	Scheme	Marks
8	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
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.

(5 marks)

Tier 3 · Hard

A town has

2400

registered sports-centre users:

960

are aged under

25

720

are aged

25

44

480

are aged

45

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 9 is 6 marks)

Mark scheme

Mark scheme for question 9
Question	Scheme	Marks
9	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
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.

(6 marks)

Q10

Tier 3 · Hard

10.

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 10 is 5 marks)

Mark scheme

Mark scheme for question 10
Question	Scheme	Marks
10	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
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.

(5 marks)

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