7 A biologist wishes to test whether the mean concentration \(\mu\), in suitable units, of a certain pollutant in a river is below the permitted level of 0.5 . She measures the concentration, \(x\), of the pollutant at 50 randomly chosen locations in the river. The results are summarised below. $$n = 50 \quad \Sigma x = 23.0 \quad \Sigma x ^ { 2 } = 13.02$$

1. Carry out a test at the \(5 \%\) significance level of the null hypothesis \(\mu = 0.5\) against the alternative hypothesis \(\mu < 0.5\).
   Later, a similar test is carried out at the \(5 \%\) significance level using another sample of size 50 and the same hypotheses as before. You should assume that the standard deviation is unchanged.
2. Given that, in fact, the value of \(\mu\) is 0.4 , find the probability of a Type II error.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.