It is known that the variable X ~ N$($population mean,$16).$

It is desired that the null hypothesis population mean$=12$ be tested against the alternative hypothesis population mean $=12$ with probability of a Type I error of $1\%.$ A random sample of $15$ observations of X is taken and the sample mean is taken to be the test statistic.

a$.$ Find the acceptance and rejection regions in terms of sample mean $.$

b$.$ For the case of population mean$=15,$ find the probability of a type II error and the power of the test