8 The random variable \(X\) has the distribution \(\mathrm { N } \left( \mu , 8 ^ { 2 } \right)\). A test is carried out, at the \(5 \%\) significance level, of \(\mathrm { H } _ { 0 } : \mu = 30\) against \(\mathrm { H } _ { 1 } : \mu > 30\), based on a random sample of size 18 .

1. Find the critical region for the test.
2. If \(\mu = 30\) and the outcome of the test is that \(\mathrm { H } _ { 0 }\) is rejected, state the type of error that is made. On a particular day this test is carried out independently a total of 20 times, and for 4 of these tests the outcome is that \(\mathrm { H } _ { 0 }\) is rejected. It is known that the value of \(\mu\) remains the same throughout these 20 tests.
3. Find the probability that \(\mathrm { H } _ { 0 }\) is rejected at least 4 times if \(\mu = 30\). Hence state whether you think that \(\mu = 30\), giving a reason.
4. Given that the probability of making an error of the type different from that stated in part (ii) is 0.4 , calculate the actual value of \(\mu\), giving your answer correct to 4 significant figures. \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}