Suppose that $[X∣p]∼Bin(15,p)$, and the goal is to test if $H_0:p=0.25$ and $H_a:p=0.75$. Suppose that $H_0$ is rejected if $X≥9$.
What is the chance of Type I error? Type II Error?
I found the p-value (type 1 Error) to be $0.00419$, but am unsure how to find the chance of type II error
On
You reject $H_0$ if the probability lies in the critical region, and in this problem that is: $X\geq 9$. The null hypothesis, $H_0$, is that $p=0.25$, and the alternate hypothesis, ${H_0}^{\small\complement}$ a.k.a. $H_a$, is that $p=0.75$.
Type I error, the false positive rate or significance level, is the probability for rejecting the null-hypothesis when it is true.
Type II error, the false negative rate, is the probability for accepting the null-hypothesis when it is false.
$$\begin{align}\tag{Type I}\alpha &= \mathsf P(\textrm{Reject }H_0\mid H_0)\\[1ex]&= \mathsf P(X\geq 9\mid p=0.25)\\[2ex]\tag{Type II}\beta&=\mathsf P(\textrm{Accept }H_0\mid {H_0}^{\small\complement})\\[1ex]&=\mathsf P(X< 9\mid p=0.75)\\[1ex]&=1-\mathsf P(X\geq 9\mid p=0.75)\end{align}$$
Congratulations; you have found $\alpha$ okay. You may find $\beta$ almost the same way.
Type I error: \begin{align*} \mathbb{P}\{\text{reject $H_0$}\mid\text{$H_0$ is true}\}&=\mathbb{P}\{X\geq 9\mid p=0.25\}\\&=\sum_{i=9}^{15}{\binom{15}{i}\cdot 0.25^i\cdot 0.75^{15-i}}\\&=0.00419. \end{align*} Similarly, Type II error: \begin{align*} \mathbb{P}\{\text{Fail to reject $H_0$}\mid\text{$H_a$ is true}\}&=\mathbb{P}\{X<9\mid p=0.75\}\\&=\sum_{i=0}^{8}{\binom{15}{i}\cdot 0.75^i\cdot 0.25^{15-i}}\\&=0.0566. \end{align*}