7 The weights, \(X\) kilograms, of bags of carrots are normally distributed. The mean of \(X\) is \(\mu\). An inspector wishes to test whether \(\mu = 2.0\). He weighs a random sample of 200 bags and his results are summarised as follows.
$$\Sigma x = 430 \quad \Sigma x ^ { 2 } = 1290$$
- Carry out the test, at the \(10 \%\) significance level.
- You may now assume that the population variance of \(X\) is 1.85 . The inspector weighs another random sample of 200 bags and carries out the same test at the \(10 \%\) significance level.
(a) State the meaning of a Type II error in this context.
(b) Given that \(\mu = 2.12\), show that the probability of a Type II error is 0.652 , correct to 3 significant figures.