# Neyman-Pearson Lemma

a quick question about the NP lemma.

So that NP Lemma is tool to find a critical region which minimizes the type $2$ error in hypothesis testing for a fixed size of the test. We usually end up with a critical region of the form $\{ \underline{x}:\overline{x} \geq c\}$ or $\{ \underline{x}:\overline{x} \leq c\}$ where $c$ is some constant related to the size of the test.

What I am interested in is, is there a quick way to determine the sign in the critical region, i.e. make a sanity check? I expect it to be related to the expectation of the distribution the random variables are from.

For example: Suppose $X_1, . . . , X_n$ is a random sample from $N(µ, σ^2)$ where the variance is known.Find the most powerful test of size α of $H_0 : µ = 0$ against $H_1 : µ = µ_1$, where $µ_1 > 0$.

The critical region will be of the form $\{ \underline{x}:\overline{x} \geq c\}$ and if we change $H_1$ to $µ_1 < 0$ then the region will be $\{ \underline{x}:\overline{x} \leq c\}$

Cheers

In an elementary case such as the one you mention, you can let intuition be your guide. You are dealing with data $\mathbf{x}$ from $\mathsf{Norm}(\mu,\sigma)$ with $\mu$ unknown and $\sigma$ known. The sample mean $\bar x$ gives information about the population mean $\mu.$
Therefore, if you are testing $H_0: \mu = 0$ vs. $H_1: \mu = \mu_1 > 0$, then you would favor $H_1$ (reject) for values of $\bar x$ sufficiently far above $0.$ This means that the critical region is $\{\mathbf{x}: \bar x > c\}$ (and you will have critical value $c > 0$).
Notes: (a) The reason I specified an elementary case is that you have to be careful to work through the logic of the N-P Lemma--especially in more advanced situations. For example, suppose data are from $\mathsf{Exp}(rate = \lambda),$ and you are testing $H_0: \lambda = 1$ vs. $H_1: \lambda = \lambda_1 > 1.$ Then the sample mean $\bar x$ gives you information about the population mean $\mu = 1/\lambda.$ Then you would reject for small values of $\bar x$ because small $\mu$ implies large $\lambda.$