5 The time $T$ seconds needed for a computer to be ready to use, from the moment it is switched on, is a normally distributed random variable with standard deviation 5 seconds. The specification of the computer says that the population mean time should be not more than 30 seconds.\\
(i) A test is carried out, at the $5 \%$ significance level, of whether the specification is being met, using the mean $\bar { t }$ of a random sample of 10 times.
\begin{enumerate}[label=(\alph*)]
\item Find the critical region for the test, in terms of $\bar { t }$.
\item Given that the population mean time is in fact 35 seconds, find the probability that the test results in a Type II error.\\
(ii) Because of system degradation and memory load, the population mean time $\mu$ seconds increases with the number of months of use, $m$. A formula for $\mu$ in terms of $m$ is $\mu = 20 + 0.6 m$. Use this formula to find the value of $m$ for which the probability that the test results in rejection of the null hypothesis is 0.5 .
\end{enumerate}

\hfill \mbox{\textit{OCR S2 2010 Q5 [11]}}