Statistics 3 (S3): SamplingEdexcel International A Level Further Maths: Topic test

1. 1
   A library has 2,400 members, each with a membership number from 0001 to 2400 on its database. A researcher wants a simple random sample of 40 members, using a table of four-digit random numbers.

   (a)

   Which of the following is the sampling frame for this survey?

   [1 mark]

   • AThe 40 members who are selected
   • BThe database listing all 2,400 members with their membership numbers
   • CThe table of four-digit random numbers
   • DThe questionnaire given to each member

   (b)

   A second researcher chooses a random start between 1 and 60 and then takes every 60th member from the database. Which statement about this sample is correct?

   [1 mark]

   • AIt is a simple random sample, because every member has the same chance of being selected
   • BIt is a stratified sample, because the members are listed in order
   • CIt is a quota sample, because the interval is fixed
   • DIt is a systematic sample, and not every possible group of 40 members can be selected

   (c)

   The first six four-digit numbers read from the table are 2431, 0875, 0000, 1990, 0875 and 2204. State which members are selected from these numbers, giving a reason for each number that is not used.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A factory produces 1,800 bottles in a shift, which pass along a conveyor belt in order. An inspector wants to check a sample of 45 bottles and chooses them systematically.

   (a)

   What is the interval between selected bottles?

   [1 mark]

   • A$40$
   • B$45$
   • C$39$
   • D$1755$

   (b)

   Which of the following is a possible weakness of this method?

   [1 mark]

   • AA random number is needed for every bottle that is selected
   • BSome bottles in the shift have no chance of being selected
   • CIf a fault occurs regularly on every 40th bottle, the sample could miss it or contain only faulty bottles
   • DThe bottles must first be divided into strata

   (c)

   The first bottle chosen is the 17th. Find the number of the last bottle chosen.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A university has 3,200 undergraduates: 1,280 study science, 1,024 study arts and 896 study business. The students' union wants a stratified sample of 100 undergraduates.

   (a)

   Calculate the number of students that should be selected from each faculty.

   [3 marks]

   (b)

   Explain why a stratified sample is preferable to a simple random sample for this survey, and describe how the 28 business students should be chosen.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A city has 48,000 adult residents: 14,400 aged 18 to 34, 16,800 aged 35 to 54 and 16,800 aged 55 or over. A polling company wants to estimate support for a new cycle lane scheme, using a sample of 600 adults.

   (a)

   Interviewers stand in the high street and approach people until quotas for each age group are filled. Describe how the quotas would be decided and how the interviewers would use them, and give two disadvantages of this quota sample compared with a stratified sample taken from a list of all adult residents.

   [6 marks]

   (b)

   The company instead uses a list of all 48,000 adult residents and takes a stratified sample by age group. Calculate the number of people to be sampled from each age group. In the 55 or over group, a systematic sample is taken from the list of 16,800 people. Find the interval between selected people and, given that the first person selected is the 33rd on the list, find the position on the list of the 5th person selected. State one circumstance in which this systematic sample could be biased.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A researcher is to carry out a quota survey of 80 shoppers at a shopping centre. Records show that half of the shoppers are women and that 40% of all shoppers are under 30. Gender and age may be assumed to be independent.

   (a)

   Which of the following is a reason for choosing quota sampling here rather than stratified sampling?

   [1 mark]

   • AEvery shopper has a known and equal chance of being selected
   • BIt allows the sampling error to be calculated
   • CNo complete list of all shoppers exists, and the results are needed quickly
   • DIt removes bias caused by the interviewer's choices

   (b)

   How many women aged 30 or over should be in the quota?

   [1 mark]

   • A$24$
   • B$16$
   • C$32$
   • D$48$

   (c)

   The researcher interviews only shoppers who are walking slowly and look willing to stop. Explain why this is likely to bias the results.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A researcher wants to find the views of all adult residents of a large town on local bus services. She uses the town's telephone directory, which lists households with a landline, as her sampling frame.

   (a)

   Which of the following is a limitation of this sampling frame?

   [1 mark]

   • AIt contains too many households to select a simple random sample
   • BIt lists the sample instead of the population
   • CIt cannot be used for systematic sampling
   • DAdults without a listed landline cannot be selected, so the frame does not cover the whole population

   (b)

   A census of the town's adults is carried out instead. Which of the following is a disadvantage of a census compared with a sample?

   [1 mark]

   • AThe results are subject to sampling error
   • BIt is likely to be costly and time-consuming for a large population
   • CIt gives a biased sample unless quotas are used
   • DIt can only be used if the population is Normally distributed

   (c)

   Suggest a better sampling frame and explain why it is better.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A forestry officer is surveying 1,350 trees in a woodland: 540 pine, 450 oak and 360 beech. A sample of 50 trees is to be selected.

   (a)

   Calculate how many trees of each species should be in a stratified sample of 50, rounding so that the total is 50.

   [3 marks]

   (b)

   The 540 pine trees are planted in 20 rows of 27 trees, numbered row by row with each row numbered from the road side. The officer takes a systematic sample of 20 pine trees from the 540. State the interval and how the first tree is chosen, and explain why this sample could be biased.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A students' union wants the views of its 12,000 members on longer library opening hours. The members are 5,400 first-years, 4,200 second-years and 2,400 final-years. A sample of 120 members is required.

   (a)

   The union is choosing between a simple random sample and a stratified sample. Calculate the numbers from each year group in a stratified sample, compare the two methods and recommend one, giving a reason.

   [6 marks]

   (b)

   Membership numbers were allocated in order of joining, so first-years have numbers 1 to 5,400, second-years 5,401 to 9,600 and final-years 9,601 to 12,000. A systematic sample of 120 members is taken from the list with a random start. Show that this sample always contains 54 first-years, 42 second-years and 24 final-years, and explain why it is not a simple random sample.

   [6 marks]

   Total for question 8: 12 marks

End of questions