Power of a test

The power of a test can be calculated with the following formula:

$${\begin{aligned}\beta _{(}\mu )&=P_{\mu }({{{\bar {x}}-\mu } \over {\sigma _{0}{/}{\sqrt {n}}}}<{{\mu _{0}-\mu } \over {\sigma _{0}{/}{\sqrt {n}}}}-{z}_{1-{{a} \over {2}}})+P_{\mu }({{{\bar {x}}-\mu } \over {\sigma _{0}{/}{\sqrt {n}}}}>{{\mu _{0}-\mu } \over {\sigma _{0}{/}{\sqrt {n}}}}+{z}_{1-{{a} \over {2}}})\\&=1-\phi ({{\mu _{0}-\mu } \over {\sigma _{0}{/}{\sqrt {n}}}}-{z}_{1-{{a} \over {2}}})+\phi ({{\mu _{0}-\mu } \over {\sigma _{0}{/}{\sqrt {n}}}}+{z}_{1-{{a} \over {2}}})\\\end{aligned}}$$

$${\begin{aligned}\beta _{(}\mu )&=P_{\mu }({{{\bar {x}}-\mu } \over {\sigma _{0}{/}{\sqrt {n}}}}<{{\mu _{0}-\mu } \over {\sigma _{0}{/}{\sqrt {n}}}}-{z}_{1-{{a} \over {2}}})+P_{\mu }({{{\bar {x}}-\mu } \over {\sigma _{0}{/}{\sqrt {n}}}}>{{\mu _{0}-\mu } \over {\sigma _{0}{/}{\sqrt {n}}}}+{z}_{1-{{a} \over {2}}})\\&=1-\phi ({{\mu _{0}-\mu } \over {\sigma _{0}{/}{\sqrt {n}}}}-{z}_{1-{{a} \over {2}}})+\phi ({{\mu _{0}-\mu } \over {\sigma _{0}{/}{\sqrt {n}}}}+{z}_{1-{{a} \over {2}}})\\\end{aligned}}$$