N2016 P2 Q6

The number of employees of a company, classified by department and gender, is shown below.

1. The directors wish to survey a sample of \(100\) of the employees. This sample is to be a stratified sample,based on department and gender.
   1. How many males should be in the sample?[1]
   2. How many females from the Development department should be in the sample?[1]

The Managing Director knows that, some years ago, the mean age of employees was \(37\) years. He believes that the mean age of employees now is less than \(37\) years.

2. State why the stratified sample from part (i) should not be used for a hypothesis test of the Managing Director's belief.[1]

The Company Secretary obtains a suitable sample of \(80\) employees in order to carry out a hypothesis test of the Managing Director’s belief that the mean age of the employees now is less than \(37\) years. You are given that the population variance of the ages is \(140\) years\(^{2}\).

3. Write down appropriate hypotheses to test the Managing Director’s belief. You are given that the result of the test, using a \(5%\) significance level, is that the Managing Director’s belief should be accepted. Determine the set of possible values of the mean age of the sample of employees.[4]
4. You are given instead that the mean age of the sample of employees is \(35.2\) years, and that the result of a test at the \(\alpha %\) significance level is that the Managing Director’s belief should not be accepted. Find the set of possible values of \(\alpha \).[3]