Sampling methods

Merchandise is sold at concerts. The manager of a concert claims that the mean value of merchandise sold to premium ticket holders is more than £6 greater than the mean value of merchandise sold to standard ticket holders.

Given that all the tickets for the next concert have been sold, describe how a stratified sample should be taken at the concert. The mean value of merchandise sold to a random sample of 60 standard ticket holders at the concert is £15 with a standard deviation of £10. The mean value of merchandise sold to a random sample of 55 premium ticket holders at the concert is £23 with a standard deviation of £8.

Test the manager’s claim at the 5% level of significance. State your hypotheses clearly.

For the test in part (b), state whether or not it is necessary to assume that values of merchandise sold have normal distributions. Give a reason for your answer.

A company director decides to survey staff about changes to the company calendar. The company has staff in 4 different job roles 72 managers, 108 drivers, 180 administrators and 360 warehouse staff. The director decides to take a stratified sample.

Write down one advantage of using a stratified sample rather than a simple random sample for this survey.

Find the number of staff in each job role that will be included in a stratified sample of 40 staff.

Describe how to choose managers for the stratified sample.

State two reasons why stratified sampling might be a more suitable sampling method than simple random sampling.

State two reasons why stratified sampling might be a more suitable sampling method than quota sampling.

A mobile library has 160 books for children on its records. The librarian believes that books with fewer pages are borrowed more often. He takes a random sample of 10 books for children.

Explain how the librarian should select this random sample.

The librarian ranked the 10 books according to how often they had been borrowed, with 1 for the book borrowed the most and 10 for the book borrowed the least. He also recorded the number of pages in each book. The results are in the table below.

Calculate Spearman’s rank correlation coefficient for these data.

Test the librarian’s belief using a 5% level of significance. State your hypotheses clearly.

A nursery has 16 staff and 40 children on its records. In preparation for an outing the manager needs an estimate of the mean weight of the people on its records and decides to take a stratified sample of size 14.

Describe how this stratified sample should be taken.

The weights, x kg, of each of the 14 people selected are summarised as

Find unbiased estimates of the mean and the variance of the weights of all the people on the nursery’s records.

Estimate the standard error of the mean.

The estimates of the standard error of the mean for the staff and for the children are 5.11 and 1.10 respectively.

Comment on these values with reference to your answer to part (c) and give a reason for any differences.

A gym club has 400 members of which 300 are males. Explain clearly how a stratified sample of size 60 could be taken.

A college manager wants to survey students’ opinions of enrichment activities. She decides to survey the students on the courses summarised in the table below.

Each student takes only one course. The manager has access to the college’s information system that holds full details of each of the enrolled students including name, address, telephone number and their course of study. She wants to compare the opinions of students on each course and has a generous budget to pay for the cost of the survey.

Give one advantage and one disadvantage of carrying out this survey using

quota sampling,

stratified sampling.

The manager decides to take a stratified sample of 100 students.

Calculate the number of students to be sampled from each course.

Describe how to choose students for the stratified sample.

A lake contains 3 species of fish. There are estimated to be 1400 trout, 600 bass and 450 pike in the lake. A survey of the health of the fish in the lake is carried out and a sample of 30 fish is chosen.

Give a reason why stratified random sampling cannot be used. (1)

State an appropriate sampling method for the survey. (1)

Give one advantage and one disadvantage of this sampling method. (2)

Explain how this sampling method could be used to select the sample of 30 fish. You must show your working. (4)

A telephone directory contains 50 000 names. A researcher wishes to select a systematic sample of 100 names from the directory.

Explain in detail how the researcher should obtain such a sample.

Give one advantage and one disadvantage of

quota sampling,

systematic sampling.