SamplingAQA A-Level Maths: Subtopic test

1. 1
   A researcher wants to estimate the mean time that students at a sixth-form college of 640 students take to travel to college. One morning she stands at the main entrance and asks the first 30 students who arrive.

   (a)

   Which sampling technique is the researcher using?

   [1 mark]

   • ASimple random sampling
   • BOpportunity sampling
   • CA census
   • DA pilot survey

   (b)

   What is the population in this investigation?

   [1 mark]

   • AThe 30 students who are asked
   • BThe travel times of the 30 students
   • CAll 640 students at the college
   • DThe students who arrive early

   (c)

   Explain why the mean travel time of her sample may give a misleading estimate of the mean for the whole college.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A teacher has an alphabetical list of the 90 students in Year 12. She numbers them 01 to 90 and uses random numbers to choose a sample of 5 students.

   (a)

   Which statement must be true for her sample to be a simple random sample?

   [1 mark]

   • AEvery student has the same chance of being selected, and every possible group of 5 is equally likely
   • BStudents with consecutive numbers are selected together
   • CThe first five students on the list are chosen
   • DEqual numbers of boys and girls are chosen

   (b)

   The random number generator gives 47, 12, 47, 93, 05, 61, 88, 30 in that order. Repeats and numbers outside 01 to 90 are rejected. Which students are selected?

   [1 mark]

   • A47, 12, 47, 93, 05
   • B12, 05, 61, 88, 30
   • C47, 12, 93, 05, 61
   • D47, 12, 05, 61, 88

   (c)

   Her sample of 5 contains 4 boys. The teacher concludes that most Year 12 students are boys. Explain why this conclusion is not justified.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A town has 24 000 registered voters. A polling company wants to estimate the proportion who support a new bypass. Method X: phone 200 voters chosen by applying random numbers to the electoral register. Method Y: ask the first 200 people who walk past a shopping centre on a Saturday morning.

   (a)

   Compare the two methods, and state which is more likely to give a representative sample of voters.

   [3 marks]

   (b)

   Using Method X, 124 of the 200 voters in a sample support the bypass. A second simple random sample of 200 voters gives 110 supporters. Use each sample to estimate the number of supporters among the 24 000 voters, and explain why the estimates differ.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A company has 2400 employees, listed in a spreadsheet and numbered 1 to 2400, with the 1500 factory workers first and the 900 office workers last. A manager wants to survey 60 employees about a new shift pattern.

   (a)

   (i) Describe how random numbers could be used to select a simple random sample of 60 employees.
   (ii) Instead, the manager asks the first 60 employees to enter the factory canteen at lunchtime. Name this technique and give one reason why the results may be biased.
   (iii) The population proportion of factory workers is $0.625$. A simple random sample of 60 contains 41 factory workers. Show that the sample proportion is higher than $0.625$ and explain whether this shows the method is biased.

   [6 marks]

   (b)

   Three independent simple random samples of 60 employees are taken. In favour of the new shift pattern: 27 in the first, 31 in the second and 22 in the third. The manager says that, because the second sample shows a majority in favour, most employees want the new pattern.
   (i) Combine the three samples to estimate the number of the 2400 employees in favour.
   (ii) Evaluate the manager's claim.

   [6 marks]

   Total for question 4: 12 marks

End of questions