Question
I have a random sample $X_1,...,X_n$ of Bernoulli$(\theta)$ random variables. Let the size of a test of a simple null $H_0:\theta=\theta_0$ against a simple alternative $H_1:\theta=\theta_1$ be $\alpha$. I found that the critical region for the test with maximum power has the form $\sum_{i=1}^{n}x_i \geq c_\alpha$ for some constant $c_\alpha$, by Neyman-Pearson.
I am not sure how to find the power function for this test, and the question actually doesn't ask for it, instead it asks for the approximate power function - ie for large samples. It says to show that the approximate power function $w(\theta)$ of the above test is given by
$w(\theta)=1-\Phi\left(\frac{\theta_0 -\theta}{\sigma(\theta)}+\frac{\sigma(\theta_0)}{\sigma(\theta)}z_\alpha\right) $
where $\Phi$ is the cdf for the standard normal, $\Phi(z_\alpha)=1-\alpha$ and $\sigma(\theta) = [\theta(1 − \theta)/n]^{\frac{1}{2}}$.
My ideas
I'm sure that from the normal cdf there must be an application of the Central Limit Theorem somewhere but I've been stuck on this for ages. Any help would be appreciated.