Null hypothesis $H_0$
Alternative hypothesis $H_1$
usually the hypothesis that we want get support by the sample data
$T = T(X_1, X_2,…,X_n)$ is a statistic to test whether $H_0$ holds.
Rejection region $W$
The range of $T$ that rejects $H_0$
$W =| \bar X -\mu_0| ≥ C$.
Acceptance region $\bar W$
Type Ⅰ error 弃真
rejects a true $H_0$.
$\alpha(C) \equiv P\{\text{rejects }H_0|H_0 \text{ is true}\} = P_{\mu = \mu_0}\{|\bar X -\mu _0|\ge C\}= p_1.$
$\Rarr p_1 = \alpha(C)= 2 -2\Phi(\frac{C}{\frac{\sigma}{\sqrt{n}}})$
Type Ⅱ error 存伪
accepts a false $H_0$.
$\beta(C) \equiv P\{\text{accepts }H_0|H_0 \text{ is false}\} =P_{\mu \ne \mu_0}\{|\bar X -\mu _0| < C\}= p_2.$
$\Rarr p_2 = \beta(C) =\Phi(\frac{\mu_0 +C -\mu}{\frac{\sigma}{\sqrt{n}}})-\Phi(\frac{\mu_0 -C -\mu}{\frac{\sigma}{\sqrt{n}}})$
For $n$ given, $\alpha(C)$ is a monotone decrease function of $C$, while $\beta(C)$ is a monotone increase function of $C$.
Therefore, it is impossible to find a limit $C$ for both $\alpha(C), \beta(C)$ to be small, as they restrict each other.
Control such that $p_1$ is no bigger than a relatively small constant $\alpha(0<\alpha<1)$, then find a hypothesis where $p_2$ is as small as possible.
Here the constant $\alpha$ is called the significance level, usually 0.01, 0.05, 0.10, etc.
Continue our discussion, take significance as $\alpha$, then $p_1 = 2 -2\Phi(\frac{C}{\frac{\sigma}{\sqrt{n}}})≤\alpha, \Phi(\frac{C}{\frac{\sigma}{\sqrt{n}}})\ge 1 - \frac{\alpha}{2}, \frac{C}{\frac{\sigma}{\sqrt{n}}}\ge z_\frac{\alpha}{2}, C\ge z_\frac{\alpha}{2}\cdot \frac{\sigma}{\sqrt{n}}.$ From Neyman-Pearson lemma, we take $C = z_\frac{\alpha}{2}\cdot \frac{\sigma}{\sqrt{n}}$ to get the smallest $p_2$.
Therefore, the rejection region $W=\{ | \bar X -\mu_0| ≥ C \}= \{| \bar X -\mu_0| ≥z_\frac{\alpha}{2}\cdot \frac{\sigma}{\sqrt{n}} \}= \{|\frac{\bar X -\mu}{\frac{\sigma}{\sqrt{n}}}|\ge z_\frac{\alpha}{2}\}$, usually written for short as $|Z| \ge z_\frac{\alpha}{2}$, which means if samples are in the rejection region(which is $|Z| ≥ z_\frac{\alpha}{2}$), then we have a certainty of $\alpha$ to reject $H_0$.
Two-sided hypothesis
$H_0: \theta =\theta_0; H_1: \theta \ne \theta_0;$