Statistics: Statistical samplingEdexcel A-Level Maths: Topic test

1. 1
   A councillor wants to find the views of the households in a ward on a proposed recycling scheme. There are $4200$ households in the ward.

   (a)

   Which of the following is a census of the households?

   [1 mark]

   • AAsking $200$ households chosen using random numbers.
   • BAsking every one of the $4200$ households.
   • CAsking the households in the street where the councillor lives.
   • DAsking every $21$st household on the electoral register.

   (b)

   Which of the following is a disadvantage of taking a census in this situation?

   [1 mark]

   • AThe result would be less accurate than a sample result.
   • BThe result would be affected by sampling variation.
   • CIt would be impossible to find the views of the whole ward.
   • DIt would be time-consuming and expensive.

   (c)

   Give one advantage and one disadvantage of the councillor using a sample rather than a census.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A leisure centre has $1500$ members. They are listed alphabetically and numbered $0001$ to $1500$. The manager wants to take a simple random sample of $60$ members to ask about the opening hours.

   (a)

   Which statement describes a simple random sample of $60$ members?

   [1 mark]

   • AEvery member has an equal chance of being chosen, and every possible group of $60$ members is equally likely.
   • BThe first $60$ members on the alphabetical list are chosen.
   • CThe $60$ members who are in the centre on Tuesday morning are chosen.
   • DMembers are chosen so that each age group is represented in proportion to its size.

   (b)

   What is the sampling frame?

   [1 mark]

   • AThe $60$ members chosen.
   • BAll the people living in the town.
   • CThe numbered list of the $1500$ members.
   • DThe random numbers used to choose the members.

   (c)

   Describe how the manager could use a random number generator to select the sample.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A hospital employs $240$ nurses, $80$ doctors and $120$ administrative staff. A manager wants to take a stratified sample of $55$ of the staff to ask about parking.

   (a)

   Calculate the number of staff of each type that should be in the sample.

   [3 marks]

   (b)

   Describe how the $30$ nurses should be selected. Explain one advantage of a stratified sample over a simple random sample of $55$ staff in this hospital.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A factory makes $20\,000$ light bulbs each day on four machines, which make $8000$, $6000$, $4000$ and $2000$ bulbs. A quality manager wants to estimate the mean lifetime of the bulbs made on one day. To find the lifetime of a bulb it must be run until it fails. The manager plans to test $100$ bulbs.

   (a)

   Explain why a census is unsuitable, and describe how the manager could select a simple random sample of $100$ bulbs. Explain why the sample mean may not equal the population mean.

   [6 marks]

   (b)

   The manager decides to take a stratified sample of $100$ bulbs by machine. Find the number of bulbs to be taken from each machine and explain why this may be better than a simple random sample. A colleague suggests a systematic sample taking every $200$th bulb off the production line instead. Describe how this would be done and give one possible source of bias.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A student wants to estimate the mean number of books borrowed each month by all the residents of her town. She stands in the town library on Monday morning and interviews the first $25$ people who walk in.

   (a)

   Which sampling method is the student using?

   [1 mark]

   • ASimple random sampling
   • BStratified sampling
   • CSystematic sampling
   • DOpportunity (convenience) sampling

   (b)

   What is the most likely source of bias in this sample?

   [1 mark]

   • APeople who visit the library borrow more books than the typical resident, so heavy borrowers are over-represented.
   • BThe sample is too large to be reliable.
   • CThe people were chosen using random numbers.
   • DThe sample was divided into too many age groups.

   (c)

   Suggest two changes that would give a more representative sample of the residents of the town.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Two researchers each take a random sample of $40$ customers from the same supermarket, on different days, and record how much each customer spent. Researcher $1$ finds a mean spend of £$32.50$. Researcher $2$ finds a mean spend of £$28.10$.

   (a)

   Which statement best explains why the two sample means are different?

   [1 mark]

   • AResearcher $2$ must have made a mistake.
   • BThe population mean must have changed between the two days.
   • CDifferent random samples from the same population will usually give different sample means.
   • DOne of the two samples must have been biased.

   (b)

   Which change would be most likely to make the sample means from different researchers more consistent?

   [1 mark]

   • ATaking a smaller random sample.
   • BTaking a larger random sample.
   • CSampling only customers who spend a lot.
   • DSampling only customers at the entrance to the shop.

   (c)

   State what the two sample means are being used to estimate, and explain why a third random sample of $40$ customers would probably give a different mean.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A warehouse stores $1200$ crates, numbered from $1$ to $1200$. An inspector wants to check a systematic sample of $30$ crates.

   (a)

   Describe how the inspector should select the sample.

   [3 marks]

   (b)

   The crates are stored in rows of $20$, numbered consecutively along each row. A fault in the packing machine affects only the last crate in each row. Explain how the systematic sample could give a biased estimate of the proportion of faulty crates.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A university has $12\,000$ students: $3600$ in Arts, $4800$ in Science, $2400$ in Engineering and $1200$ in Business. The students' union wants the views of the students on a new timetable and plans to survey $500$ of them. One officer proposes standing in the cafeteria at lunchtime and asking the first $500$ students she meets.

   (a)

   Name the sampling method proposed by the officer. Give one advantage of this method and explain why it may not give a representative sample. State what type of sample would be better and why.

   [6 marks]

   (b)

   The union decides to use a stratified sample of $500$ students by faculty. Calculate the number of students from each faculty. Describe how the Business students should be selected, and give one reason why two different stratified samples might still give different conclusions about the whole university.

   [6 marks]

   Total for question 8: 12 marks

End of questions