Question on Type I and Type II Errors for a Test of Binomial $p$ A random variable $Y$ representing the number of successes can be modeled by a binomial distribution with parameters $n=250$ and $p,$ whose value is unknown. A significance test is performed, based on a sample value $Y,$ to test the hypothesis $p=0.6$ against the alternative hypothesis $p>0.6.$ The probability of Type I error is $0.05.$
a. Find the critical region for $Y$.
b. Find the probability of making a Type II error in the case when in actual fact $p=0.675.$
 A: Here are some good clues to get you started.  According to the null hypothesis you have
$Y \sim Binom(n = 250, p = 0.6).$ You will want to reject
for large values of $Y.$ So you want to find $k$ such that
$P(Y \ge k | n=250, p=.6) \approx .05.$ Because the binomial
distribution is discrete, you probably won't get an exact
match to .05. 
By whatever method, you will find that
$P(Y \ge 164) \approx .040$ and $P(Y \ge 163) \approx .052.$
When the question asks for Type I error (significance level)
5% with a discrete distribution, it means to get as near
to 5% as possible without exceeding 5%. So it seems your
critical region (rejection region) is $\{Y \ge 164\}.$
Below is a figure in which some probabilities in the
distribution $Binom(250, .6)$ are plotted. The rejection
region is to the right of the dotted vertical line. The thin blue
curve is the density function of $Norm(\mu = 150, \sigma=7.75).$ 

Maybe you can show how you would go about verifying this
much. (My $guess$ is you are supposed to use the normal approximation
to the binomial distribution. The 'approximation' part means
that your answers may differ slightly from my exact computations
using software.) 
$If$ you do that, someone may be able to help you with the Type II
part of the question (perhaps me in the morning or someone else later
tonight). That would be $P(Y \le 163 | n=250, p=.675),$
which turns out to be near .24.
