Critique of sampling methods

2 A club has 264 members, numbered from 1 to 264 . Donash wants to choose a random sample of members for a survey. In order to choose the members for the sample he uses his calculator to generate random digits. His first 20 random digits are as follows. $$\begin{array} { l l l l } 10612 & 11801 & 21473 & 22759 \end{array}$$

1. The numbers of the first two members in the sample are 106 and 121. Write down the numbers of the next two members in the sample.
2. To obtain the numbers for members after the 4th member, Donash starts with the second random digit, 0 , and obtains the numbers 061 and 211. Explain why this method will not produce a random sample.

1 Alan wishes to choose one child at random from the eleven children in his music class. The children are numbered \(2,3,4\), and so on, up to 12 . Alan then throws two fair dice, each numbered from 1 to 6 , and chooses the child whose number is the sum of the scores on the two dice.

1. Explain why this is an unsatisfactory method of choosing a child.
2. Describe briefly a satisfactory method of choosing a child.

2 The editor of a magazine wishes to obtain the views of a random sample of readers about the future of the magazine.

1. A sub-editor proposes that they include in one issue of the magazine a questionnaire for readers to complete and return. Give two reasons why the readers who return the questionnaire would not form a random sample. The editor decides to use a table of random numbers to select a random sample of 50 readers from the 7302 regular readers. These regular readers are numbered from 1 to 7302 . The first few random numbers which the editor obtains from the table are as follows. $$49757 \quad 80239 \quad 52038 \quad 60882$$
2. Use these random numbers to select the first three members in the sample.

1 It is desired to obtain a random sample of 15 pupils from a large school. One pupil suggests listing all the pupils in the school in alphabetical order and choosing the first 15 names on the list.

1. Explain why this method is unsatisfactory.
2. Suggest a better method.

1 It is required to select a random sample of 30 pupils from a school with 853 pupils. A student suggests the following method.
"Give each pupil sequentially a three-digit number from 001 to 853 . Use a calculator to generate random three-digit numbers from 0.000 to 0.999 inclusive, multiply the answer by 853 , add 1 and round off to the nearest whole number. Select the corresponding pupil, and repeat as necessary".

1. Determine which pupil would be picked for each of the following calculator outputs: $$0.103 , \quad 0.104 , \quad 0.105 , \quad 0.106 , \quad 0.107$$
2. Use your answers to part (i) to show that this method is biased, and suggest an improvement.

3 A company has been commissioned to make 50 very expensive titanium components.
A sample of the components needs to be tested to ensure that they are sufficiently strong. However, this is a test to destruction, so the components which are tested can no longer be used.

1. Explain why it would not be appropriate to use a census in these circumstances. A manager suggests that the first 5 components to be manufactured should be tested.
2. Explain why this would not be a sensible method of selecting the sample. A statistician advises the manager that the sample selected should be a random sample.
3. Give two desirable features (other than randomness) that the sample should have.

2 A company manufactures batches of twenty thousand tins which are subsequently filled with fruit. The company tests tins from each batch to make sure that they are strong enough. The test is easy and cheap to carry out, but when a tin has been tested it is no longer suitable for filling with fruit.

1. 1. Explain why a sample size of 5 tins per batch may not be appropriate in this case.
   2. Explain why a sample size of 1000 tins per batch may not be appropriate in this case. The company tests a sample of 30 tins from each batch.
2. Explain why it would not be sensible for the sample to consist of the final 30 tins produced in a batch.
3. Give two features that the sample should have.