Power Function of a test

 

어떤 가설 $H_0:\theta =thet{a}_0$에 대하여 p-value가 $\alpha$보다 작으면 statistically significant하다고 한다.

 

 

Power Function for two-sided z-test

$$\begin{array}{rl}Power& =P_{\mu }(\text{p-value}<\alpha )\\ & =P(\left|\frac{\bar{X}-{\mu }_0}{{\sigma }_0/\sqrt{n}}\right|>z_{1-\frac{\alpha }{2}})\\ & =P(\bar{X}<{\mu }_0-z_{1-\alpha /2}\frac{{\sigma }_0}{\sqrt{n}}\text{ or }\bar{X}>{\mu }_0+z_{1-\alpha /2}\frac{{\sigma }_0}{\sqrt{n}})\\ & =1-\mathrm{\Phi }(\frac{{\mu }_0-\mu }{{\sigma }_0/\sqrt{n}}+z_{(1-\alpha /2)})+\mathrm{\Phi }(\frac{{\mu }_0-\mu }{{\sigma }_0/\sqrt{n}}-z_{(1-\alpha /2)})\end{array}$$

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.