The management of a factory thinks that the mean time required to complete a particular task is 22 minutes. The times, in minutes, taken by employees to complete this task have a normal distribution with mean $\mu $ and standard deviation 3.5. An employee claims that 22 minutes is not long enough for the task. In order to investigate this claim, the times for a random sample of 12 employees are used to test the null hypothesis $\mu  = 22$ against the alternative hypothesis $\mu  \gt 22$ at the 5% significance level.

a) Show that the null hypothesis is rejected in favour of the alternative hypothesis if $\overline x  \gt 23.7$ (correct to 3 significant figures), where $\overline x $ is the sample mean.

b) Find the probability of a Type II error given that the actual mean time is 25.8 minutes.