N2014 P2 Q9

The lateness, in minutes, of the \(0815\) bus is normally distributed with mean \(4.3\). A new company takes over the service. A random sample of \(10\) days is chosen and the lateness of the \(0815\) bus is recorded. The sample mean is \(\bar t\) minutes and the sample variance is \(k^2\) minutes squared. A test at the \(10\%\) level of significance is to be carried out to determine whether the mean lateness has been reduced.

1. State the null and alternative hypotheses, defining any symbol you use.[2]
2. Given that \(k^2=3.2\), find the set of values of \(\bar t\) for which the null hypothesis is not rejected.[4]
3. Given that \(\bar t=4\), find the set of values of \(k^2\) for which the null hypothesis is rejected.[3]