i recently started to do brush up on my high school probability, and ran across a question i could not solve. The gist of the problem goes as follows: I am given a probability of 5% that some product is defective, and am asked to test the Null-Hypothesis $H_0: p \leq 0.05$ against the Hypothesis $H_1: p> 0.05$ in a one sided hypothesis test. Error Probability is given as $\alpha = 0.1$.
The question then goes on and gives me n=250, and gives k = 18, such that Hypothesis $H_0$ is disregard for $k \geq 18$. I am now asked to find the range in which the true fraction of defective parts can lie, such that the probability of/for the type - II Error is less than $0.25$.
So far, i tried to solve the question by doing trial and error, effectively guessing the value of p necessary and calculating the type-II probability this way. What i would like to know is whether or not i am missing some relation with type-II errors, such that i can either calculate the true value of p, or the interval of defective parts necessary, such that the probability is less than $0.25$