In the context of hypothesis testing define the following terms: (i) simple hypothesis; (ii) critical region; (iii) size; (iv) power; and (v) type II error probability.

State, without proof, the Neyman-Pearson lemma.

Let $X$ be a single observation from a probability density function $f$. It is desired to test the hypothesis

$$H_{0}: f=f_{0} \quad \text { against } \quad H_{1}: f=f_{1},$$

with $f_{0}(x)=\frac{1}{2}|x| e^{-x^{2} / 2}$ and $f_{1}(x)=\Phi^{\prime}(x),-\infty<x<\infty$, where $\Phi(x)$ is the distribution function of the standard normal, $N(0,1)$.

Determine the best test of size $\alpha$, where $0<\alpha<1$, and express its power in terms of $\Phi$ and $\alpha$.

Find the size of the test that minimizes the sum of the error probabilities. Explain your reasoning carefully.