Distinguish the target population and invert a non-rejection condition

In this question you should state clearly the variables and the values of the parameters of any distribution you use.

A pharmaceutical company has 300 employees from the research department and 330 employees from the manufacturing department.

(i) Billy wishes to find out about the stress level of employees in the research department, so he sends a survey to the 300 employees in the research department. Explain whether these 300 employees form a sample or a population.1 mark

(ii) Charlie wishes to conduct a survey regarding the employees’ experiences with a new facility.100 of them are randomly selected from the research department and the other 100 are randomly selected from the manufacturing department. Explain whether the method will form a random sample.2 marks

The pharmaceutical company produces pills against a particular bacteria. The manager claims that the average time taken to manufacture a batch of pills is less than 12 hours. The time taken in hours, $x$, was measured on 80 occasions and the results are summarised as follows:

$\sum(x-12)=-12.8,\qquad\sum(x-12)^2=16.19.$

(iii) Find the unbiased estimates of the population mean and variance of the time taken to manufacture a batch of pills.2 marks

(iv) Carry out a hypothesis test on the manager’s claim. Find the p-value for this test and explain what it indicates about the manager’s claim.4 marks

It was found that the testing process had not been rigorous enough and the same hypothesis test was conducted in another random sample of 80 observations. The sample mean was found to be $k$ hours and the sample standard deviation was found to be 1.5 hours. It was found that the manager’s claim was not supported at 1% level of significance.

(v) Find the range of values of $k$, correct to 3 decimal places.3 marks