# Neyman-Pearson lemma and hypothesis test

Let $X_1...X_n$ be idndependent and identically distributed as Bernoulli random quantities with parameter $p$. Let $p_1>p_0$ and consider the test of hypotheses $H_0:p=p_0$ versus $H_1:p=p_1$

Using the Neyman-Pearson lemma, a critical region of the form $C^* = \{(x_1,...,x_n): \sum_i_=_1 ^n x_i\ge c\}$ is obtained to perform this test where $c$ is the critical value chosen so that $P(\sum_i_=_i ^n X_i \ge c|H_0$ true)=$\alpha$ and the sampling distribution of $\sum_i_=_1^n X_i$ is $Bin (n,p)$ that is Binomial with $n$ trials and probability of success $p$.

Could you please explain to me in detail the properties of the test based on the critical region $C^*$ and in particular those derived as a consequence of using the Neyman-Pearson lemma??