3 An employer has commissioned an opinion polling organisation to undertake a survey of the attitudes of staff to proposed changes in the pension scheme. The staff are categorised as management, professional and administrative, and it is thought that there might be considerable differences of opinion between the categories. There are 60,140 and 300 staff respectively in the categories. The budget for the survey allows for a sample of 40 members of staff to be selected for in-depth interviews.

1. Explain why it would be unwise to select a simple random sample from all the staff.
2. Discuss whether it would be sensible to consider systematic sampling.
3. What are the advantages of stratified sampling in this situation?
4. State the sample sizes in each category if stratified sampling with as nearly as possible proportional allocation is used. The opinion polling organisation needs to estimate the average wealth of staff in the categories, in terms of property, savings, investments and so on. In a random sample of 11 professional staff, the sample mean is \(\pounds 345818\) and the sample standard deviation is \(\pounds 69241\).
5. Assuming the underlying population is Normally distributed, test at the \(5 \%\) level of significance the null hypothesis that the population mean is \(\pounds 300000\) against the alternative hypothesis that it is greater than \(\pounds 300000\). Provide also a two-sided \(95 \%\) confidence interval for the population mean.
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