The question is as follows:
Consider an iid sample $\{X_1, \cdots, X_5\}$ out of tossing a coin 5 times. Each $X_i$ takes 0 or 1 depending on whether $i$th toss is head or tail, respectively. Let $p$ denote the probability of the toss being tail.
(a) construct a test for the null hypothesis of $p=0.5$ against the alternative hypothesis of $p>0.5$ and justify your test.
(b) Now suppose the sample size has increased to 200. How would you construct your test?
So to solve (a), I want to use likelihood ratio test. I thought that $X_1, \cdots, X_5$ follows $B(5,p)$. And I derived that the mle of $p$ is $\bar{X}$. So, to set a lambda as
$\Lambda=\dfrac{\prod \binom{5}{x_i} \hat{p}^{\sum x_{i}}(1-\hat{p})^{5-\sum x_i}} {\prod \binom{5}{x_i} (0.5)^{\sum x_{i}}(1-0.5)^{5-\sum x_i}}=\dfrac{\bar{x}^{\sum x_i} (1-\bar{x})^{5-\sum{x_i}}}{(\frac{1}{2})^5} =32\bar{x}^{\sum x_i} (1-\bar{x})^{5-\sum{x_i}}=32\bar{x}^{5\bar{x}} (1-\bar{x})^{5(1-\bar{x})} $
I don't know how to make progress here.. How can I dervie critical region using that?