6 The weight of a plastic box manufactured by a company is \(W\) grams, where \(W \sim \mathrm {~N} ( \mu , 20.25 )\). A significance test of the null hypothesis \(\mathrm { H } _ { 0 } : \mu = 50.0\), against the alternative hypothesis \(\mathrm { H } _ { 1 } : \mu \neq 50.0\), is carried out at the \(5 \%\) significance level, based on a sample of size \(n\).

1. Given that \(n = 81\),
   (a) find the critical region for the test, in terms of the sample mean \(\bar { W }\),
   (b) find the probability that the test results in a Type II error when \(\mu = 50.2\).
2. State how the probability of this Type II error would change if \(n\) were greater than 81 .