N2011 P2 Q10

In a factory, the time in minutes for an employee to install an electronic component is a normally distributed continuous random variable \(T\). The standard deviation of \(T\) is \(5.0\) and under ordinary conditions the expected value of \(T\) is \(38.0\). After background music is introduced into the factory, a sample of \(n\) components is taken and the mean time taken for randomly chosen employees to install them is found to be \(\bar t\) minutes. A test is carried out, at the \(5\%\) significance level, to determine whether the mean time taken to install a component has been reduced.

1. State appropriate hypotheses for the test, defining any symbols you use.[2]
2. Given that \(n=50\), state the set of values of \(\bar t\) for which the result of the test would be to reject the null hypothesis.[3]
3. It is given instead that \(\bar t=37.1\) and the result of the test is that the null hypothesis is not rejected. Obtain an inequality involving \(n\), and hence find the set of values that \(n\) can take.[4]