Test function

ϕ (x_{1}, \dots, x_{n}) = {\begin{cases} 1 if (x_{1}, \dots, x_{n}) \in C, \\ 0 if (x_{1}, \dots, x_{n}) \in C^{c} \end{cases}

If $ϕ (x_{1}, \dots, x_{n}) = 1$ , reject $H_{0}$
If $ϕ (x_{1}, \dots, x_{n}) = 0$ , do not reject $H_{0}$

Example

$ϕ (x_{1}, \dots, x_{n}) \sim B e r n o u l l i (p)$ , $E (ϕ (x_{1}, \dots, x_{n})) = p$

\begin{aligned} α & = P (reject H_{0} | H_{0} is true) \\ = P (ϕ (x_{1}, \dots, x_{n}) \in C | H_{0} is true) \\ = E (ϕ (\underset{\sim}{x}) | H_{0} is true) \\ β & = P (do not reject H_{0} | H_{0} is false) \\ = P (ϕ (x_{1}, \dots, x_{n}) \in C^{c} | H_{0} is false) \\ = 1 - E (ϕ (\underset{\sim}{x}) | H_{0} is false) \end{aligned}