# Neyman Pearson Lemma

I've been reading up on Neyman Pearson Lemma but don't understand it to its full extent. Could someone please explain to me how to obtain the most powerful size in a bernoulli distribution?

Given a random sample $$(X_1, \ldots, X_n)$$ from a $$\mathrm{Bernoulli}(p)$$ distribution, the Neyman-Pearson Lemma tests the null hypothesis $$H_0 : p = p_0$$ against the alternative $$H_1 : p = p_1$$. The test rejects $$H_0$$ if $$\frac{L(p_0)}{L(p_1)}=\frac{\prod_{i=1}^np_0^{x_i}(1-p_0)^{1-x_i}}{\prod_{i=1}^np_1^{x_i}(1-p_1)^{1-x_i}}=\frac{p_0^{\sum_{i=1}^n x_i}(1-p_0)^{n-\sum_{i=1}^n x_i}}{p_1^{\sum_{i=1}^n x_i}(1-p_1)^{n-\sum_{i=1}^n x_i}} for some $$k$$. This condition can also be written as $$\left(\frac{p_0(1-p_1)}{p_1(1-p_0)}\right)^{\sum_{i=1}^n x_i} for some $$k'$$. Thus, when $$p_0 < p_1$$, we must have $$\sum_{i=1}^n x_i > c$$ for some $$c$$ and when $$p_0 > p_1$$,we must have $$\sum_{i=1}^n x_i < c$$ for some $$c$$.