Sampling method explanation

7 A market researcher is investigating the length of time that customers spend at an information desk. He plans to choose a sample of 50 customers on a particular day.

1. He considers choosing the first 50 customers who visit the information desk. Explain why this method is unsuitable.
   The actual lengths of time, in minutes, that customers spend at the information desk may be assumed to have mean \(\mu\) and variance 4.8. The researcher knows that in the past the value of \(\mu\) was 6.0. He wishes to test, at the \(2 \%\) significance level, whether this is still true. He chooses a random sample of 50 customers and notes how long they each spend at the information desk.
2. State the probability of making a Type I error and explain what is meant by a Type I error in this context.
3. Given that the mean time spent at the information desk by the 50 customers is 6.8 minutes, carry out the test.
4. Give a reason why it was necessary to use the Central Limit theorem in your answer to part (c).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 The mean breaking strength of cables made at a certain factory is supposed to be 5 tonnes. The quality control department wishes to test whether the mean breaking strength of cables made by a particular machine is actually less than it should be. They take a random sample of 60 cables. For each cable they find the breaking strength by gradually increasing the tension in the cable and noting the tension when the cable breaks.

1. Give a reason why it is necessary to take a sample rather then testing all the cables produced by the machine.
2. The mean breaking strength of the 60 cables in the sample is found to be 4.95 tonnes. Given that the population standard deviation of breaking strengths is 0.15 tonnes, test at the \(1 \%\) significance level whether the population mean breaking strength is less than it should be.
3. Explain whether it was necessary to use the Central Limit theorem in the solution to part (ii).

7

1. Give a reason why sampling would be required in order to reach a conclusion about
   1. the mean height of adult males in England,
   2. the mean weight that can be supported by a single cable of a certain type without the cable breaking.
2. The weights, in kg , of sacks of potatoes are represented by the random variable \(X\) with mean \(\mu\) and standard deviation \(\sigma\). The weights of a random sample of 500 sacks of potatoes are found and the results are summarised below. $$n = 500 , \quad \Sigma x = 9850 , \quad \Sigma x ^ { 2 } = 194125 .$$
   1. Calculate unbiased estimates of \(\mu\) and \(\sigma ^ { 2 }\).
   2. A further random sample of 60 sacks of potatoes is taken. Using your values from part (b) (i), find the probability that the mean weight of this sample exceeds 19.73 kg .
   3. Explain whether it was necessary to use the Central Limit Theorem in your calculation in part (b) (ii).

The manager of a leisure club is considering a change to the club rules. The club has a large membership and the manager wants to take the views of the members into consideration before deciding whether or not to make the change.

1. Explain briefly why the manager might prefer to use a sample survey rather than a census to obtain the views. [2]
2. Suggest a suitable sampling frame. [1]
3. Identify the sampling units. [1]