Exercise :
For the estimation of the unknown rate of votes $p$ that a political group $A$ will gather in the following elections, suppose we selected a random sample of $n=15$ voters. Suppose that you want to check the null hypothesis $H_0 : p = 0.5$ against the alternative $H_1 : p< 0.5$. Suppose that the critical region of this specific hypothesis test is $K=\{y\leq 2\}$, where $y$ is the observed number of voters who voted for the political group $A$.
i) Calculate the confidence level $a$ of the above hypothesis test.
ii) If the political group $A$ eventually gathers a rate of $30\%$ in the elections, calculate the probability of the type II error for the above hypothesis test.
Question - Request : For our upcoming exams, we are supposed to be able to handle such problems but we weren't really introduced to any of them due to shortage of time. I kindly request some tips or a thorough explanation such as I will be able to get a grip on such problems. (This is a past exams exercise)
So $\alpha$ represents the probability of making a Type I error, that is, rejecting $H_0$ when $H_0$ is true, that is why we say "we are 95% confident that $H_0$ is true", because there is a $1-95\%=\alpha$ probability of this conclusion not being true. Therefore:
\begin{align} \alpha&=Pr(\text{Reject } H_0 | H_0) \\ \\ &=Pr(y \leq 2 | p=0.5) \\ \\ &= \sum_{i=0}^2 {15\choose{i}}0.5^i(1-0.5)^{15-i} \\ \\ &=0.5^{15}\sum_{i=0}^2 {15\choose{i}} \\ \end{align}
Now $\beta$ is the probability of doing a Type II error given $p=0.3$:
\begin{align} \beta&=Pr(\text{Accept }H_0|H_1) \\ \\ &=Pr(y > 2 | p=0.3) \\ \\ &=1-Pr(y \leq 2 | p=0.3) \\ \\ &=1-\sum_{i=0}^2{15\choose{i}}0.3^i(1-0.3)^{15-i} \end{align}