2021 RI P2 Q6

The recruitment manager of the private car hire company, I-ber, claims that the mean weekly earnings of a full-time driver is \(\$980\). The managing director suspects that the mean weekly earnings is less than \(\$980\) and he instructs the recruitment manager to carry out a hypothesis test on a sample of drivers. It is given that the population standard deviation of the weekly earnings is \(\$88\).

1. State suitable hypotheses for the test, defining any symbols that you use.[2]

The recruitment manager takes a random sample of \(10\) drivers. He finds that the weekly earnings in dollars, are as follows.

\(942 \quad 950 \quad 905 \quad 1003 \quad 883 \quad 987 \quad 924 \quad 920 \quad 913 \quad 968\)

2. Find the mean weekly earnings of the sample of these \(10\) drivers. Carry out the test, at \(5\%\) level of significance, for the recruitment manager. Give your conclusion in context and state a necessary assumption for the test to be valid.[5]
3. Find the smallest level of significance at which the test would result in rejection of the null hypothesis, giving your answer correct to \(1\) decimal place.[1]