Sampling frames and populations

1 A magazine conducted a survey about the sleeping time of adults. A random sample of 12 adults was chosen from the adults travelling to work on a train.

1. Give a reason why this is an unsatisfactory sample for the purposes of the survey.
2. State a population for which this sample would be satisfactory. A satisfactory sample of 12 adults gave numbers of hours of sleep as shown below. \(4.6 \quad 6.8\) 5.2
   6.2
   5.7 \(\begin{array} { l l } 7.1 & 6.3 \end{array}\) 5.6
   7.0 \(5.8 \quad 6.5\) 7.2
3. Calculate unbiased estimates of the mean and variance of the sleeping times of adults.

1. (a) State one characteristic of a population that would make a census a practical alternative to sampling.

A leisure centre has 2500 members.
It asks a sample of 300 members for their opinions on the fees it charges for using the centre. For the sample,
(b) (i) identify a suitable sampling frame,
(ii) identify a sampling unit. The leisure centre has the following pieces of information. \(A\) is the list of the different types of membership that can be paid for by members. \(B\) is the mean of the membership fees paid by all 2500 members. \(C\) is the number in the sample of 300 members who are satisfied with the fees they pay.
(c) State the piece of information that is a statistic. Give a reason for your answer.

2. (a) Explain what you understand by (i) a population and (ii) a sampling frame. The population and the sampling frame may not be the same.
(b) Explain why this might be the case.
(c) Give an example, justifying your choices, to illustrate when you might use

1. a census,
2. a sample.

A large dental practice wishes to investigate the level of satisfaction of its patients.

1. Suggest a suitable sampling frame for the investigation. [1]
2. Identify the sampling units. [1]
3. State one advantage and one disadvantage of using a sample survey rather than a census. [2]
4. Suggest a problem that might arise with the sampling frame when selecting patients. [1]

A large dental practice wishes to investigate the level of satisfaction of its patients.

1. Suggest a suitable sampling frame for the investigation. [1]
2. Identify the sampling units. [1]
3. State one advantage and one disadvantage of using a sample survey rather than a census. [2]
4. Suggest a problem that might arise with the sampling frame when selecting patients. [1]

A random sample \(X_1, X_2, \ldots, X_n\) is taken from a finite population. A statistic \(Y\) is based on this sample.

1. Explain what you understand by the statistic \(Y\). [2]
2. Give an example of a statistic. [1]
3. Explain what you understand by the sampling distribution of \(Y\). [2]

Explain briefly what you understand by

1. a sampling frame, [1]
2. a statistic. [2]

Before introducing a new rule the secretary of a golf club decided to find out how members might react to this rule.

1. Explain why the secretary decided to take a random sample of club members rather than ask all the members. [1]
2. Suggest a suitable sampling frame. [1]
3. Identify the sampling units. [1]

Explain what you understand by

1. a population, [1]
2. a statistic. [1]

A researcher took a sample of 100 voters from a certain town and asked them who they would vote for in an election. The proportion who said they would vote for Dr Smith was 35\%.

1. State the population and the statistic in this case. [2]
2. Explain what you understand by the sampling distribution of this statistic. [1]

An insurance company is investigating how often its customers crash their cars.

1. Suggest an appropriate sampling frame. [1 mark]
2. Describe the sampling units. [1 mark]
3. State the advantage of a sample survey over a census in this case. [2 marks]

A company that makes ropes for mountaineering wants to assess the breaking strain of its ropes.

1. Explain why a sample survey, and not a census, should be used. [2 marks]
2. Suggest an appropriate sampling frame. [1 mark]