Statistics: Sampling - Worksheets, Questions and Revision

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A-Level · Statistics

S1 Statistics: Sampling

EDEXCEL 9MA0 · Calculator allowed · about 95 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

A rail company wants to investigate the average delay time, in minutes, experienced by passengers across all 2.4 million passenger journeys it operated last year. The company holds a complete computer record of the delay time for every one of these journeys.

(a)State what is meant by a census.(1)

(b)The company could analyse the delay time for all 2.4 million journeys, i.e. take a census. Give one advantage of doing this rather than using a sample.(1)

(c)Give one disadvantage of the company using a census in this situation.(1)

(d)The company instead decides to analyse a sample of 500 journeys. Give one reason why using a sample could be preferable to a census here.(1)

(Total for Question 1 is 4 marks)

For each statement below about using a sample rather than a census, state whether the statement describes an advantage or a disadvantage of sampling.

(a)"Results can usually be obtained more quickly."(1)

(b)"The results may be less accurate, since conclusions about the whole population are based on only part of it."(1)

(c)"Less time and money are usually needed to collect and process the data."(1)

(d)"No individual data is available for population members who were not chosen, so no prediction can be made about any one of them specifically."(1)

(Total for Question 2 is 4 marks)

Priya is a student researcher who wants to survey students at her large sixth-form college about their travel-to-college habits. She obtains a printed list of all students enrolled on A-level and BTEC courses from the college's admissions database, dated 1 September. She plans to carry out her survey the following March.

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

(b)Explain one way in which the admissions database, printed on 1 September, might not be a suitable sampling frame for Priya's survey in March.(2)

(c)Suggest one improvement Priya could make to obtain a more suitable sampling frame.(1)

(Total for Question 3 is 4 marks)

A stable owner has 84 horses, numbered from 00 to 83. She wants to select a simple random sample of 4 horses to check for a particular health condition.

(a)Describe fully how the owner could use a random number generator to select this simple random sample, ensuring no horse is chosen more than once.(3)

(b)The random digits below were generated and grouped into pairs, reading from left to right: 51 84 07 07 62 39. Given that horses are numbered 00 to 83, identify the 4 horses selected by this method.(2)

(Total for Question 4 is 5 marks)

A market researcher stands outside a shopping centre. Based on past data, she knows that 40% of shoppers at the centre are men and 60% are women. She plans to stop and interview shoppers about a new energy drink until she has interviewed 20 men and 30 women.

(a)State the name of this sampling method.(1)

(b)Give one advantage of this method compared with using stratified sampling in this situation.(1)

(c)Give one disadvantage of this method.(1)

(d)Suggest one way the researcher could reduce the bias identified in part (c).(1)

(Total for Question 5 is 4 marks)

A local council wants to find out residents' opinions on a proposed new cycle lane. It posts a survey link on the council's official social media page, and asks residents to comment with their views.

(a)State the sampling method used.(1)

(b)Explain two separate ways in which this method could lead to a biased sample of residents' opinions.(2)

(c)Suggest a more appropriate sampling method the council could use, given that it holds the electoral register for the area.(1)

(Total for Question 6 is 4 marks)

A supermarket had 850 customers pass through its tills between 9 a.m. and 5 p.m. last Saturday. The store manager wants to select a systematic sample of 25 of these customers to survey about checkout waiting times.

(a)Calculate the sampling interval, k, that the manager should use.(2)

(b)Describe fully how the manager should select the systematic sample of 25 customers, using the value of k found in part (a).(2)

(c)The manager considers only sampling customers who pass through the tills between 9 a.m. and 10 a.m. each day, to make identifying the sample quicker. Explain why this could introduce bias into the results.(2)

(Total for Question 7 is 6 marks)

A university researcher is investigating the exercise habits of the 18000 students at a large university, who live across 12 halls of residence.

(a)The researcher plans to survey the first 50 students who arrive at the university gym one Monday morning. State the name of this sampling method.(1)

(b)Explain one disadvantage of the method in part (a) for this particular study.(2)

(c)As an alternative, the researcher randomly selects 3 of the 12 halls of residence and surveys every student living in those 3 halls. State the name of this sampling method, and give one advantage of using it instead of the method in part (a).(2)

(Total for Question 8 is 5 marks)

A sixth-form college has 840 students: 460 in Year 12 and 380 in Year 13. The college wants to take a stratified sample of 60 students, by year group, to survey about wellbeing support.

(a)Show that the number of Year 13 students required in the sample is 27, to the nearest whole number.(2)

(b)Find the number of Year 12 students required in the sample.(2)

(c)Once the number of students required from Year 12 has been found, explain how the college should select which Year 12 students to include.(1)

(Total for Question 9 is 5 marks)

A large secondary school has 1500 students across Years 7 to 13. The head teacher wants to survey students about the quality of school lunches. The school's admissions system holds a complete, up to date list of every student, together with their year group and name.

(a)Give one reason why taking a sample, rather than a census of all 1500 students, would be appropriate here.(1)

(b)Recommend a suitable sampling method for the head teacher to use, and justify your choice with reference to the information given.(3)

(c)State one limitation of the method you recommended in part (b).(1)

(Total for Question 10 is 5 marks)

A scientist records the daily maximum temperature at a weather station for every day of one calendar year (365 days): 90 days in Winter, 92 days in Spring, 92 days in Summer and 91 days in Autumn. She wants to select a sample of 20 days for detailed analysis, with the number of days chosen from each season in proportion to the number of days in that season.

(a)State the name of the sampling method described.(1)

(b)Calculate the number of days that should be sampled from Summer.(2)

(c)Calculate the number of days that should be sampled from Winter, giving your answer to the nearest whole number.(2)

(d)Explain why, when using stratified sampling with proportional allocation, rounding each stratum's calculated sample size independently can sometimes cause a problem, and state how this can be resolved.(2)

(Total for Question 11 is 7 marks)

A gym has 3 types of membership: Standard (240 members), Premium (160 members) and Family (100 members). A stratified sample of 25 members, proportional to membership type, is taken to survey satisfaction scores out of 10.

(a)Calculate the number of Premium members that should be included in the sample.(2)

(b)The satisfaction scores of the 8 Premium members sampled are: 7, 9, 6, 8, 10, 7, 9, 8. Calculate the mean satisfaction score for this stratum.(2)

(c)The mean satisfaction scores for the Standard and Family strata are 6.5 (from the 12 Standard members sampled) and 7.6 (from the 5 Family members sampled) respectively. Calculate an estimate of the overall mean satisfaction score across the whole sample of 25 members.(3)

(Total for Question 12 is 7 marks)

A factory production line manufactures 1350 identical components during a shift. The quality control team wants a systematic sample of 40 components.

(a)Calculate the sampling interval, giving your answer to the nearest whole number.(2)

(b)Using an interval of 34, explain why selecting every 34th component from a random start will result in fewer than 40 components being sampled during the shift, and determine how many components will actually be sampled.(3)

(c)Suggest one way the quality control team could adjust their method to obtain the full sample of exactly 40 components.(1)

(Total for Question 13 is 6 marks)

A firm employs 300 staff across three departments: Sales (120 staff: 72 male, 48 female), Production (150 staff: 90 male, 60 female) and Admin (30 staff: 12 male, 18 female). Human Resources wants a stratified sample of 40 staff, proportional to department size.

(a)Show that the number of staff sampled from the Production department is 20.(2)

(b)Find the number of staff that should be sampled from the Sales department and from the Admin department.(3)

(c)Within the Sales department, 72 of the 120 staff are male and 48 are female. Using the Sales sample size found in part (b), calculate how many male and how many female staff should be included in the Sales sample, rounding appropriately so that your two values sum to 16.(3)

(Total for Question 14 is 8 marks)

Mark scheme · S1 Statistics: Sampling

Question 1

(a) B1 a survey/study that collects data from every member of the population, oe
(a) Answer: A survey or study that collects data from every member of the population.
(b) B1 gives completely accurate results with no sampling error, since the whole population is used, oe
(b) Answer: It gives a completely accurate result, with no sampling error, since every journey is included.
(c) B1 processing/analysing 2.4 million records is far more time-consuming (and costly) than using a sample, oe
(c) Answer: Processing 2.4 million individual records is far more time-consuming and expensive than analysing a sample.
(d) B1 results can be obtained much more quickly, since far fewer records need to be processed, oe (must be a different point from part (c))
(d) Answer: Results can be produced much more quickly, since only 500 records need to be processed rather than 2.4 million.

Question 2

(a) B1 advantage
(a) Answer: Advantage.
(b) B1 disadvantage
(b) Answer: Disadvantage.
(c) B1 advantage
(c) Answer: Advantage.
(d) B1 disadvantage
(d) Answer: Disadvantage.

Question 3

(a) B1 a list of all the members of the population, from which the sample is selected, oe
(a) Answer: A list of all the members of the population, used to select the sample.
(b) B1 identifies that some students will have left their course and new students will have joined between September and March, so the list is out of date
(b) B1 explains the consequence: some students currently at the college have no chance of selection, and some listed students may no longer attend, so the frame does not accurately represent the population being studied in March, oe
(b) Answer: The list is six months out of date, so some students who have since left cannot actually be surveyed, and new students who joined after September have no chance of being selected, meaning the frame does not accurately represent the population in March.
(c) B1 request an updated/current enrolment list from the college, printed close to the date of the survey, oe
(c) Answer: Ask the college for an up to date enrolment list, printed close to March when the survey will actually take place.

Question 4

(a) B1 assigns each horse a unique two-digit number from 00 to 83, using the existing list as the sampling frame
(a) B1 uses a random number generator to produce two-digit random numbers, rejecting any outside the range 00 to 83
(a) B1 also rejects any repeated numbers, continuing until 4 different horses have been selected
(a) Answer: Label the horses 00 to 83. Generate random two-digit numbers, ignoring any greater than 83 and any repeats, until 4 different horses have been chosen.
(b) M1 correctly rejects 84 (outside the range 00 to 83) and the second 07 (a repeat)
(b) A1 cao: horses 51, 07, 62 and 39 selected, in this order
(b) Answer: Horses 51, 07, 62 and 39.

Question 5

(a) B1 quota sampling
(a) Answer: Quota sampling.
(b) B1 no sampling frame of all shoppers is needed, and it is quicker/cheaper to carry out on location, oe
(b) Answer: No sampling frame of every shopper is required, so it is quicker and cheaper to carry out on the spot than stratified sampling.
(c) B1 the researcher chooses who to approach, so the sample is non-random and interviewer bias can be introduced, e.g. avoiding people who look busy or unfriendly, oe
(c) Answer: The researcher personally chooses who to approach, so the sample is non-random and interviewer bias can occur, for example avoiding people who look busy.
(d) B1 approach every nth shopper who passes, rather than personally choosing who to stop, oe
(d) Answer: Approach every nth shopper who walks past, rather than personally selecting who to stop.

Question 6

(a) B1 self-selecting sample (volunteer sample)
(a) Answer: Self-selecting sample (volunteer sample).
(b) B1 only residents who use/follow that particular social media platform can respond, so residents without an account or who are less active online, e.g. some older residents, are excluded
(b) B1 residents with strong opinions, for or against the cycle lane, are more likely to respond than those who feel neutral, so the sample may over-represent extreme views, oe
(b) Answer: Residents who do not use that social media platform, such as some older residents, cannot respond at all; and residents with strong feelings, for or against the cycle lane, are more likely to comment than those who are neutral, so the results may over-represent extreme opinions.
(c) B1 take a simple random sample or a stratified sample of residents from the electoral register, oe
(c) Answer: Take a simple random sample (or a stratified sample) of residents directly from the electoral register.

Question 7

(a) M1 850 / 25
(a) A1 cao k = 34
(a) Answer: k = 34
(b) B1 choose a random starting point between 1 and k (i.e. between 1 and 34)
(b) B1 then select every 34th customer after that point (e.g. by adding 34 repeatedly), until 25 customers have been chosen
(b) Answer: Pick a random start between 1 and 34, then select every 34th customer after that until 25 customers are chosen.
(c) B1 customers shopping between 9 and 10 a.m. may differ systematically from customers shopping at other times, e.g. more likely to be doing a quick shop before work
(c) B1 so the sample would not represent all types of customer, and conclusions about waiting times could not be generalised to the whole day, oe
(c) Answer: Early-morning shoppers may differ from customers at other times of day (for example, doing a quick shop before work), so the sample would not represent all types of customer and results could not be generalised to the whole day.

Question 8

(a) B1 opportunity sampling (convenience sampling)
(a) Answer: Opportunity sampling (convenience sampling).
(b) B1 only students who already choose to use the gym can be included in the sample
(b) B1 so students with different exercise habits, e.g. those who never exercise or who exercise elsewhere, have no chance of selection, which is likely to bias the results, oe
(b) Answer: Only students who already attend the gym can be sampled, so students who never exercise or who exercise elsewhere have no chance of selection, biasing the results towards more active students.
(c) B1 cluster sampling
(c) B1 advantage, e.g. quicker/cheaper to carry out since students are grouped geographically, and every type of student (not just gym-users) has a chance of selection, oe
(c) Answer: Cluster sampling. It is quicker and cheaper to carry out, since students are grouped geographically in halls, and it includes all types of student, not just those who already use the gym.

Question 9

(a) M1 380 / 840 x 60
(a) A1 cso: 27.14... which rounds to 27 (answer given, exact working must be shown)
(a) Answer: 27 (given)
(b) M1 460 / 840 x 60, or 60 - 27
(b) A1 cao 33
(b) Answer: 33 students.
(c) B1 use simple random sampling (or systematic sampling) to choose 33 students at random from the Year 12 group, so each Year 12 student has an equal chance of selection, oe
(c) Answer: Use simple random sampling to choose 33 students at random from within Year 12, so every Year 12 student has an equal chance of being selected.

Question 10

(a) B1 much quicker and cheaper to survey a sample than every one of the 1500 students, and a well-chosen sample can still give a reliable estimate of opinion, oe
(a) Answer: A sample is much quicker and cheaper to survey than all 1500 students, and a well-chosen sample can still give a reliable estimate of opinion.
(b) B1 names an appropriate method, e.g. stratified sampling by year group
(b) B1 justifies using the sampling frame available, which records each student's year group, so the number of students per year can be found and used to calculate proportional strata sizes
(b) B1 further valid justification, e.g. stratified sampling ensures every year group is represented in proportion to its size, avoiding a year group being over- or under-represented as could happen with simple random sampling, oe
(b) Answer: Stratified sampling by year group is recommended, since the school's list records each student's year group, allowing proportional strata sizes to be calculated; this also guarantees every year group is fairly represented, unlike simple random sampling, which could by chance over- or under-represent a year group.
(c) B1 it takes more time and is more complex to set up than simple random sampling, since students must first be split into strata and each stratum's sample size calculated, oe
(c) Answer: It takes more time and effort to set up than simple random sampling, since students must first be divided into strata and each stratum's sample size calculated separately.

Question 11

(a) B1 stratified sampling
(a) Answer: Stratified sampling.
(b) M1 92 / 365 x 20
(b) A1 awrt 5 days
(b) Answer: 5 days (to the nearest whole number).
(c) M1 90 / 365 x 20
(c) A1 awrt 5 days
(c) Answer: 5 days.
(d) B1 the individually rounded values may not sum to the required overall sample size, e.g. they could sum to one more or one fewer than needed
(d) B1 one (or more) of the values must then be adjusted, typically the value(s) closest to rounding the other way, so that the strata sizes sum to the required total, oe
(d) Answer: Rounding each stratum's value separately may mean the totals do not add up to the required sample size, e.g. 20; this is resolved by adjusting one of the values (usually the one closest to rounding the other way) until the totals sum correctly.

Question 12

(a) M1 160 / 500 x 25
(a) A1 cao 8
(a) Answer: 8 Premium members.
(b) M1 sum of the 8 scores divided by 8
(b) A1 cao 8
(b) Answer: Mean = 8.
(c) M1 12 x 6.5 + 8 x 8 + 5 x 7.6
(c) M1 = 78 + 64 + 38 = 180
(c) A1 cao 180 / 25 = 7.2
(c) Answer: Overall mean = 7.2.

Question 13

(a) M1 1350 / 40 (= 33.75)
(a) A1 cao 34
(a) Answer: Interval = 34.
(b) B1 34 x 40 = 1360, which is greater than 1350, so there are not enough components in the shift to make 40 complete selections at this interval
(b) M1 1350 / 34
(b) A1 cao: 1350/34 = 39.7..., so only 39 components can be selected
(b) Answer: Only 39 components can be sampled.
(c) B1 reduce the sampling interval to 33 (since 1350/40 = 33.75 rounded down), or treat the production line as circular and wrap around to the start once the end is reached, oe
(c) Answer: Use a smaller interval of 33 instead of 34 (or treat the list of components as circular, wrapping back to the start once the end of the shift is reached).