Suppose n $=12$ independent Bernoulli$(\backslash$pi $)$ trials $($where $\backslash$pi is the probability of success$)$ are conducted. Suppose we wish to test the hypotheses

H$0$ :$\backslash$pi $=0\mathrm{.}6$ vs$.$ H$1$ :$\backslash$pi $>0\mathrm{.}6,$

and we wish to use X$,$ the number of successes obtained in the $12$ trials, as a test statistic. Then $($using binomial tables$)$:

$1$

$($a$)$ Explain what would be a reasonable critical region and why?

$($b$)$ If we choose X in $\{0,1,2\},$ X in $\{0,1,2,3\},$ X in $\{10,11,12\}$ and X in $\{9,10,11,12\}$ to be four critical regions, find the corresponding probabilities of Type$-$I error, and comment;

$($c$)$ When the true value of $\backslash$pi $=0\mathrm{.}8($i$.$e$.,$ H$1$ is true$),$ find the probabilities of Type$-$II error for the above four critical regions, and comment;

$($d$)$ What are the corresponding values of power for the four critical regions, and what do you observe from them?