Sampling Methods Worksheets, Questions and Revision

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GCSE · Statistics

S06 Sampling Methods

EDEXCEL 1ST0 · Calculator allowed · about 80 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

1

For each statement about sampling, write down whether it is True or False.

(i)A sample is a small part of a population, used to find out information about the whole population.(1)

(ii)A random sample is one where every member of the population has an equal chance of being chosen.(1)

(iii)A convenience sample, chosen because its members are easy to reach, always gives an unbiased picture of the whole population.(1)

(iv)A census means collecting data from every single member of the population.(1)

(Total for Question 1 is 4 marks)

2

A student wants to find out the favourite lunch food of everyone in her school. She only asks the friends sitting with her at lunch.

(a)Name the sampling method she has used.(1)

(b)Give one disadvantage of using this method to find out about the lunch preferences of the whole school.(2)

(Total for Question 2 is 3 marks)

3

Ridgeway Council wants to find the mean number of people living in each household in a town with 12000 households.

(a)Write down the term used for collecting data from every household in the population, rather than from a sample of them.(1)

(b)Give one advantage of the council surveying a sample of 300 households instead of taking a census.(1)

(c)Give one advantage of taking a census instead of a sample in this situation.(1)

(d)Give one disadvantage of the council taking a census of all 12000 households.(1)

(Total for Question 3 is 4 marks)

4

Cranmoor Tennis Club has 60 members, numbered 1 to 60 in the membership register. The club wants to select a simple random sample of 8 members to ask about a new clubhouse.

(a)Describe a method the club could use to select a simple random sample of 8 members.(3)

(b)Explain what is meant by a random sample.(2)

(Total for Question 4 is 5 marks)

5

A market researcher stops shoppers outside a supermarket until she has interviewed 15 men and 15 women, whichever shoppers happen to walk past.

(a)Name the sampling method described.(1)

(b)Give two reasons why this method may not give a sample that represents all shoppers at the supermarket.(2)

(Total for Question 5 is 3 marks)

6

A factory numbers 500 items leaving the production line in order, from 1 to 500. An inspector picks a random starting point between 1 and 10, then inspects every 10th item after that.

(a)Name the sampling method described.(1)

(b)The inspector's random starting point is item 6. Write down the item numbers of the first four items she will inspect.(2)

(c)The factory wants to inspect exactly 50 items using this method, out of the 500 produced. Work out the sampling interval needed.(2)

(Total for Question 6 is 5 marks)

7

Fenwick Electronics has 3400 loyalty card holders, each with a unique membership number from 0001 to 3400. The company wants a simple random sample of 20 card holders.

(a)Describe how the company could use a random number generator to select this sample.(3)

(b)Give one advantage of using a random number generator instead of a person picking numbers 'at random' by eye.(1)

(Total for Question 7 is 4 marks)

8

A phone manufacturer takes a random sample of 5% of the 8000 phones made in a batch, to test for faults.

(a)Work out how many phones are in this 5% sample.(2)

(b)In the sample, 6 phones were found to be faulty. Use this to estimate the number of faulty phones in the whole batch of 8000.(2)

(Total for Question 8 is 4 marks)

9

A school has 1200 students. A random sample of 150 students is surveyed about how they travel to school. In the sample, 36 students say they cycle to school.

(a)Work out the proportion of the sample who cycle to school.(1)

(b)Use your answer to part (a) to estimate the total number of students at the school who cycle to school.(2)

(Total for Question 9 is 3 marks)

10

An office building has staff working across 4 floors. The bar chart shows the number of staff working on each floor.

(a)Write down the number of staff working on Floor 3.(1)

(b)Work out the total number of staff working in the whole building.(2)

(c)Describe a suitable method for choosing a sample of staff that includes people from every floor of the building, rather than just one floor.(2)

(d)Give one problem that could occur if the manager only samples staff from Floor 1.(1)

(Total for Question 10 is 6 marks)

11

A sports centre with 2000 members wants to sample its members. Which of these is an example of systematic sampling?

A) Choosing every member whose surname begins with the letter S
B) Selecting the 30th member on the membership list, then every 30th member after that
C) Asking members who happen to be at the gym on Monday morning
D) Using a computer program to pick 50 completely random membership numbers

(Total for Question 11 is 1 mark)

12

A council plans to survey residents of a town using names and addresses taken from the electoral register.

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

(b)Give one reason why the electoral register might not be a suitable sampling frame for finding the opinions of all town residents.(2)

(c)Explain how using a poor sampling frame could lead to bias in the survey's results.(2)

(Total for Question 12 is 5 marks)

13

The table below compares three sampling methods.

Method	Advantage	Disadvantage
Simple random sampling	Every member of the population has an equal chance of being chosen	(a)
Systematic sampling	Quick and easy to select once the sampling interval is fixed	(b)
Stratified sampling	Every group in the population is represented in proportion to its size	(c)

(a)State one disadvantage of simple random sampling.(1)

(b)State one disadvantage of systematic sampling.(1)

(c)State one disadvantage of stratified sampling.(1)

(d)A company's 500 staff are split unevenly across 5 very different-sized departments. State which of the three sampling methods above would be most suitable, and give a reason.(2)

(Total for Question 13 is 5 marks)

14

Millbrook Secondary School has 900 students split across three year groups. The bar chart shows the number of students in each year group. The school wants to take a stratified sample of 45 students across the three year groups.

(a)State the formula used to work out how many students to sample from a given year group in stratified sampling.(1)

(b)Work out the number of Year 9 students that should be in the sample.(2)

(c)Work out the number of Year 10 students that should be in the sample.(2)

(d)Given that the Year 11 sample size is 14, show that the three sample sizes together total 45.(1)

(Total for Question 14 is 6 marks)

15

A student wants to investigate whether Year 10 students (180 students) spend more time on homework each week than Year 11 students (160 students) at her school. She plans to take a stratified sample of 34 students across the two year groups.

(a)Plan: Give one reason why a stratified sample of both year groups is more appropriate than only surveying Year 10 students.(1)

(b)Collect: The student wants an overall sample of 34 students. Work out how many Year 10 students and how many Year 11 students should be in the sample.(3)

(c)Show that these two sample sizes total 34.(1)

(d)Process: Describe how the student should choose which specific 18 Year 10 students to survey.(1)

(e)Interpret: The mean weekly homework time was 5.2 hours for the Year 10 sample and 6.8 hours for the Year 11 sample. Give one reason, other than a genuine difference between the year groups, why these means might differ.(1)

(Total for Question 15 is 7 marks)

16

A researcher wants a systematic sample of 1 in 20 from a list of 600 customers, numbered 1 to 600 in the order that they joined a loyalty scheme (customer 1 joined first, customer 600 joined most recently).

(a)Work out how many customers will be in this systematic sample.(2)

(b)The researcher wants to know whether newer customers spend differently from customers who joined a long time ago. Explain why this systematic sample, based on join order, is well suited to answering that question.(2)

(c)Give one general situation in which systematic sampling could produce a biased sample.(2)

(Total for Question 16 is 6 marks)

17

A charity has 2500 registered supporters. It wants to estimate what percentage of all its supporters would be willing to volunteer at an upcoming event, using a simple random sample of 4% of supporters.

(a)Work out how many supporters are in this 4% sample.(2)

(b)Describe a suitable method, using random numbers, for selecting this simple random sample from the charity's full list of 2500 supporters.(3)

(c)In the sample, 28 supporters said they would be willing to volunteer. Use this to estimate the percentage of all 2500 supporters who would be willing to volunteer.(2)

(d)Give one reason why this estimate might not exactly match the true percentage for all 2500 supporters.(1)

(Total for Question 17 is 8 marks)

Mark scheme · S06 Sampling Methods

Question 1

(i) B1 True cao
(i) Answer: True
(ii) B1 True cao
(ii) Answer: True
(iii) B1 False cao
(iii) Answer: False
(iv) B1 True cao
(iv) Answer: True

Question 2

(a) B1 Convenience sampling cao
(a) Answer: Convenience sampling
(b) B1 identifies that her friends may not represent the tastes of the whole school, oe
(b) B1 explains that friendship groups often share similar tastes/backgrounds, so the results could be biased towards her own circle, oe
(b) Answer: Her friends are unlikely to represent the whole school; friendship groups often share similar tastes, so the results may be biased.

Question 3

(a) B1 Census cao
(a) Answer: Census
(b) B1 a sample is quicker and cheaper to carry out and process than surveying all 12000 households, oe
(b) Answer: A sample of 300 is much quicker and cheaper to carry out than contacting all 12000 households.
(c) B1 a census gives completely accurate results for the whole population, with no sampling error, oe
(c) Answer: A census gives an exact result for the whole population, with no sampling error.
(d) B1 collecting and processing data from all 12000 households takes much longer and costs much more than a sample, oe
(d) Answer: It would take a very long time and be expensive to contact and process data from all 12000 households.

Question 4

(a) B1 uses the existing numbers 1 to 60 already assigned to members, oe
(a) B1 a genuine random method is used to generate the numbers, e.g. a lottery draw from identical tickets or a random number generator, oe
(a) B1 continues until 8 different members are chosen, ignoring any repeated numbers, oe
(a) Answer: Use the numbers 1 to 60 already given to members; generate random numbers (e.g. with a random number generator or by drawing tickets from a bag) and select the matching members, ignoring repeats, until 8 different members have been chosen.
(b) B1 every member of the population has an equal chance of being selected, oe
(b) B1 each member is selected independently of any other, oe
(b) Answer: A random sample is one where every member of the population has an equal chance of being chosen, independently of any other member.

Question 5

(a) B1 Quota sampling cao (accept opportunity sampling)
(a) Answer: Quota sampling
(b) B1 only shoppers visiting at that particular time and day are available to be chosen, so other shoppers have no chance of selection, oe
(b) B1 the researcher may unconsciously choose people who look easier or more approachable to stop, introducing personal bias, oe
(b) Answer: Only shoppers present at that time and day can be chosen, and the researcher may unconsciously pick people who seem more approachable, both of which can bias the sample.

Question 6

(a) B1 Systematic sampling cao
(a) Answer: Systematic sampling
(b) M1 adds the interval of 10 repeatedly starting from 6, oe
(b) A1 6, 16, 26, 36 cao
(b) Answer: 6, 16, 26, 36
(c) M1 500 / 50, oe
(c) A1 10 cao
(c) Answer: 10

Question 7

(a) B1 generates random numbers between 0001 and 3400, oe
(a) B1 matches each generated number to the card holder with that membership number, oe
(a) B1 ignores any repeated numbers and continues until 20 different card holders are chosen, oe
(a) Answer: Use a random number generator to produce numbers between 0001 and 3400, select the card holder matching each number, ignore any repeats, and continue until 20 different card holders have been chosen.
(b) B1 removes human bias/judgement from the selection, so the sample is genuinely random, oe
(b) Answer: It removes human bias, since a person choosing 'at random' often unintentionally favours certain numbers.

Question 8

(a) M1 0.05 x 8000, oe
(a) A1 400 cao
(a) Answer: 400
(b) M1 6/400 x 8000, oe
(b) A1 120 cao
(b) Answer: 120

Question 9

(a) B1 36/150 oe (0.24, 24%, 6/25) cao
(a) Answer: 36/150 = 0.24
(b) M1 0.24 x 1200, oe
(b) A1 288 cao
(b) Answer: 288

Question 10

(a) B1 20 cao
(a) Answer: 20
(b) M1 15+30+20+15, oe
(b) A1 80 cao
(b) Answer: 80
(c) B1 the number of staff chosen from each floor reflects how many staff work on that floor, oe
(c) B1 within each floor, the individuals are chosen using a random method, oe
(c) Answer: Choose a number of staff from each floor in proportion to how many work there, then select those individuals at random within each floor.
(d) B1 staff on Floors 2, 3 and 4 have no chance of being selected, so their views/characteristics are missed and the sample is unrepresentative, oe
(d) Answer: Staff on the other three floors would have no chance of being included, so the sample would not represent the whole building.

Question 11

B1 B cao
Answer: B) Selecting the 30th member on the membership list, then every 30th member after that

Question 12

(a) B1 a list of all the members of the population from which a sample can be drawn, oe
(a) Answer: A sampling frame is a list of every member of the population that a sample can be selected from.
(b) B1 identifies that the electoral register excludes people who are not registered to vote, oe
(b) B1 gives an example, e.g. residents under 18 or residents who have not registered, so these groups have no chance of being sampled, oe
(b) Answer: It excludes residents who are not registered to vote, such as those under 18, so their opinions could never be included in the sample.
(c) B1 if certain groups of the population are missing from the frame entirely, they have no chance of being selected, oe
(c) B1 so the sample, and its results, may systematically differ from/not represent the true population, oe
(c) Answer: If whole groups are left out of the frame, they can never be chosen, so the sample - and any conclusions drawn from it - may not truly represent the whole population.

Question 13

(a) B1 a complete, up-to-date list of the whole population (a sampling frame) is needed before sampling can start, and this can be impractical for very large populations, oe
(a) Answer: A complete list of the whole population is needed first, which can be impractical for very large populations.
(b) B1 if there is a hidden pattern in the list that matches the sampling interval, the sample could be biased, oe
(b) Answer: If the list has a hidden repeating pattern that matches the interval used, the sample could be biased.
(c) B1 the population must be accurately divided into groups (strata) before sampling, which takes extra time and information to organise, oe
(c) Answer: The population must first be accurately split into groups, which takes extra time and information to arrange.
(d) B1 Stratified sampling
(d) B1 because it guarantees every department, however small, is represented in the sample in proportion to its size, oe
(d) Answer: Stratified sampling, because it makes sure every department - even a small one - is represented fairly in proportion to its size.

Question 14

(a) B1 (number of students in the year group / total number of students) x overall sample size, oe
(a) Answer: (stratum size / population size) x sample size
(b) M1 320/900 x 45, oe
(b) A1 16 cao
(b) Answer: 16
(c) M1 300/900 x 45, oe
(c) A1 15 cao
(c) Answer: 15
(d) B1 16 + 15 + 14 = 45
(d) Answer: 16 + 15 + 14 = 45

Question 15

(a) B1 surveying only Year 10 gives no data at all for Year 11, so no fair comparison between the two year groups is possible, oe
(a) Answer: Surveying only Year 10 would give no information about Year 11 at all, making a fair comparison impossible.
(b) M1 34/340 (=0.1) used as the sampling fraction, oe
(b) A1 Year 10: 18 cao
(b) B1 Year 11: 16 cao
(b) Answer: Year 10: 18, Year 11: 16
(c) B1 18 + 16 = 34
(c) Answer: 18 + 16 = 34
(d) B1 use a random method, e.g. a random number generator applied to the Year 10 register, to choose 18 different students, oe
(d) Answer: Use a random method, such as random numbers applied to the Year 10 register, to select 18 different students.
(e) B1 sampling variation - a different random sample from each year group could have given different mean values, oe
(e) Answer: Sampling variation: a different random sample from each year group could easily have produced different mean values.

Question 16

(a) M1 600/20, oe
(a) A1 30 cao
(a) Answer: 30
(b) B1 the customers are already ordered by when they joined, oe
(b) B1 so sampling at a regular interval along this list spreads the sample evenly across old and new customers, giving a fair mix of join dates, oe
(b) Answer: Because the list is already ordered by join date, sampling at a fixed interval along it spreads the sample evenly across old and new customers.
(c) B1 identifies that the population list has a repeating pattern, oe
(c) B1 and this pattern matches, or is a multiple of, the sampling interval, so the same type of member is repeatedly chosen or repeatedly missed, oe
(c) Answer: If the list has a repeating pattern that lines up with the sampling interval (for example, always landing on the first customer served each week), the sample could repeatedly pick up, or repeatedly miss, a particular type of member.

Question 17

(a) M1 0.04 x 2500, oe
(a) A1 100 cao
(a) Answer: 100
(b) B1 assigns each of the 2500 supporters a unique number (e.g. 1 to 2500) from the full list, oe
(b) B1 uses a random number generator to generate numbers within that range, oe
(b) B1 selects the matching supporters, ignoring repeats, until 100 different supporters have been chosen, oe
(b) Answer: Number the 2500 supporters 1 to 2500; use a random number generator to produce numbers in that range; select the matching supporters, ignoring repeats, until 100 different supporters have been chosen.
(c) M1 28/100 x 100, oe
(c) A1 28% cao
(c) Answer: 28%
(d) B1 sampling variation - a different random sample of 100 supporters could give a different percentage, oe
(d) Answer: Sampling variation: a different random sample of 100 supporters could easily give a different percentage.

Sampling Methods - Worksheets, Questions and Revision

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