Hypothesis test for a Normal mean

The idea

To test a mean, use the distribution of the sample mean: if $X \sim N(\mu, \sigma^2)$, then $\bar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right)$. Assume $H_0$'s mean, find the probability of a sample mean at least as extreme, and compare with the level.

1. Write $H_0: \mu = \dots$ and $H_1$.
2. Under $H_0$, $\bar{X} \sim N\left(\mu_0, \frac{\sigma^2}{n}\right)$; standard error $\frac{\sigma}{\sqrt{n}}$.
3. Find the $p$-value or the critical value and conclude in context.

Watch out Use $\frac{\sigma}{\sqrt{n}}$ for the sample mean, not $\sigma$.