Power Function of a test

어떤 가설 $H_0: \theta = theta_0$에 대하여 p-value가 $\alpha$보다 작으면 statistically significant하다고 한다.

 

 

Power Function for two-sided z-test

\begin{align*} Power &= P_{\mu}(\text{p-value} < \alpha) \\ &= P\left( \left| \cfrac{\overline{X} - \mu_0}{\sigma_0 / \sqrt{n}} \right| > z_{1-\frac{\alpha}{2}} \right) \\ &= P\left( \overline{X} < \mu_0 - z_{1-\alpha/2}\cfrac{\sigma_0}{\sqrt{n}} \text{ or } \overline{X} > \mu_0 + z_{1-\alpha/2}\cfrac{\sigma_0}{\sqrt{n}} \right) \\ &= 1 - \Phi\left( \cfrac{\mu_0 - \mu}{\sigma_0 / \sqrt{n}} + z_{(1-\alpha/2)} \right) + \Phi\left( \cfrac{\mu_0 - \mu}{\sigma_0 / \sqrt{n}} - z_{(1-\alpha/2)} \right) \end{align*}

Power Function of z-test about $\mu$

만약 $\mu=\mu_0$라면 power=$\alpha$가 되고, $\mu$가 $\mu_0$로부터 멀어질수록 power는 $1$에 가까워진다.