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 $p=0.675.$
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.