The recruitment manager of the private car hire company, I-ber, claims that the mean weckly earnings of a full-time driver is \(\$ 980\). The managing director suspects that the mean weekly earmings 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 weckly earnings is \(\$ 88\).

(i) 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 carnings in dollars, are as follows.

$\begin{array}{llllllllll}942 & 950 & 905 & 1003 & 883 & 987 & 924 & 920 & 913 & 968\end{array}$

(ii) Find the mean weekly earnings of the sample of these 10 drivers. Carry out the test, at (5 %) Ievel of significance, for the recruitment manager. Give your conclusion in context and state a necessary assumption for the test to be valid.

[5]

(iii) 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]

(i) Null hypothesis, $H_0: \mu=980$

Alternative hypothesis, $H_1: \mu<980$

where $\mu$ is the population mean weekly salary.

(ii) $\bar{x}=939.5$

$p$-value $=0.0728>0.05$, hence we do not reject $H_0$, and conclude that, based on the test carried out by the recruitment manager, there is insufficient evidence for the managing director to conclude at $5 \%$ level of significance that the mean weekly earnings of a driver is less than $\$ 980$.

(iii)$7.3 \%$ (1 d.p)