La región de rechazo una vez establecido un nivel de significación \(\alpha\) se puede expresar de dos formas alternativas,
\[ \begin{aligned} R_{c(\alpha)} & =\left\{\underset{\sim}{x} \in \mathbb{R}^{n}: \frac{\mathcal{L}\left(\mu_{1} ; \underset{\sim}{x}\right)}{\mathcal{L}\left(\mu_{0} ; \underset{\sim}{x}\right)}>c(\alpha)\right\} \\ & =\left\{\underset{\sim}{x} \in \mathbb{R}^{n}: \frac{\mathcal{L}\left(\mu_{0} ; \underset{\sim}{x}\right)}{\mathcal{L}\left(\mu_{1} ; \underset{\sim}{x}\right)}<c^{*}(\alpha)\right\} \end{aligned} \]
donde \(c^{*}(\alpha)=1 / c(\alpha)\) y
\[ \begin{aligned} \mathcal{L}(\mu ; \underset{\sim}{x}) & =\prod_{i=1}^{n}\left(2 \pi \sigma_{0}\right)^{-\frac{1}{2}} \exp \left\{-\frac{1}{2 \sigma_{0}^{2}}\left(x_{i}-\mu\right)^{2}\right\} \\ & =\left(2 \pi \sigma_{0}\right)^{-\frac{n}{2}} \exp \left\{-\frac{1}{2 \sigma_{0}^{2}} \sum_{i=1}^{n}\left(x_{i}-\mu\right)^{2}\right\}. \end{aligned} \]
Desarrollando el cuadrado del exponente del numerador y del denominador, obtenemos
\[ \begin{aligned} \frac{\mathcal{L}\left(\mu_{1} ; \underset{\sim}{x}\right)}{\mathcal{L}\left(\mu_{0} ; \underset{\sim}{x}\right)} & =\exp \left\{-\frac{1}{2 \sigma_{0}^{2}}\left(\sum_{i=1}^{n}\left(x_{i}-\mu_{0}\right)^{2}-\sum_{i=1}^{n}\left(x_{i}-\mu_{1}\right)^{2}\right)\right\} \\ & =\exp \left\{\frac{n}{2 \sigma_{0}^{2}}\left(2 \bar{x}_{n}\left(\mu_{1}-\mu_{0}\right)-\left(\mu_{1}^{2}-\mu_{0}^{2}\right)\right)\right\}, \end{aligned} \]
donde \(\mu_{1}-\mu_{0}>0\) y
\[ \begin{aligned} R_{c(\alpha)} & =\left\{\underset{\sim}{x} \in \mathbb{R}^{n}: \bar{x}_{n}>c^{\prime}(\alpha)\right\} \\ c^{\prime}(\alpha) & =\frac{1}{\mu_{1}-\mu_{0}}\left[\frac{1}{n} \sigma_{0}^{2} \log c(\alpha)+\frac{1}{2}\left(\mu_{1}^{2}-\mu_{0}^{2}\right)\right]. \end{aligned} \]
Al ser \(\bar{X}_{n} \sim \mathcal{N}\left(\mu, \frac{\sigma_{0}^{2}}{n}\right)\) continua no se requiere aleatorizar en la frontera de la región de rechazo y se tiene la siguiente función test
\[ \phi(\underset{\sim}{x})= \begin{cases}1, & \underset{\sim}{x} \in R_{c(\alpha)} \\ 0, & \underset{\sim}{x} \notin R_{c(\alpha)}\end{cases}. \]
A partir de ella (o directamente a partir de la región de rechazo) se puede calcular la cota de la región de rechazo,
\[ \begin{aligned} \alpha & =E\left[\phi(\underset{\sim}{X}) ; \mu_{0}\right]=P\left(\phi(\underset{\sim}{X})=1 ; \mu_{0}\right) \\ & =P\left(\underset{\sim}{X} \in R_{c(\alpha)} ; \mu_{0}\right)=P\left(\sqrt{n} \frac{\bar{X}_{n}-\mu_{0}}{\sigma_{0}}>c^{\prime \prime}(\alpha)\right) \\ & =P\left(\mathcal{N}(0,1)>c^{\prime \prime}(\alpha)\right), \\ c^{\prime \prime}(\alpha) & =\sqrt{n} \frac{c^{\prime}(\alpha)-\mu_{0}}{\sigma_{0}}=z_{\alpha} \end{aligned} \]
de modo que
\[ \begin{aligned} R_{c(\alpha)} & =\left\{\underset{\sim}{x} \in \mathbb{R}^{n}: \sqrt{n} \frac{\bar{x}_{n}-\mu_{0}}{\sigma_{0}}>z_{\alpha}\right\} \\ & =\left\{\underset{\sim}{x} \in \mathbb{R}^{n}: \bar{x}_{n}>\mu_{0}+z_{\alpha} \frac{\sigma_{0}}{\sqrt{n}}\right\} . \end{aligned} \]
Para calcular la función de potencia en \(\mu\) repetimos el mismo esquema que para \(\alpha\) pero bajo el supuesto general de que \(\mu \in \Theta=\left\{\mu_{0}, \mu_{1}\right\}\) :
\[ \begin{aligned} \beta(\mu) & =E[\phi(\underset{\sim}{X}) ; \mu]=P(\phi(\underset{\sim}{X})=1 ; \mu) \\ & =P\left(\underset{\sim}{X} \in R_{c(\alpha)} ; \mu\right)=P\left(\bar{X}_{n}>\mu_{0}+z_{\alpha} \frac{\sigma_{0}}{\sqrt{n}} ; \mu\right) \\ & =P\left(\sqrt{n} \frac{\bar{X}_{n}-\mu}{\sigma_{0}}>\sqrt{n} \frac{\mu_{0}-\mu}{\sigma_{0}}+z_{\alpha}\right) \\ & =1-\Phi\left(\sqrt{n} \frac{\mu_{0}-\mu}{\sigma_{0}}+z_{\alpha}\right) \end{aligned} \]
Propiedades de la función de potencia en este ejemplo:
\[ \lim _{\mu_{1} \rightarrow+\infty} \beta\left(\mu_{1}\right)=1-\lim _{\mu_{1} \rightarrow+\infty} \Phi\left(\sqrt{n} \frac{\mu_{0}-\mu_{1}}{\sigma_{0}}+z_{\alpha}\right)=1-\Phi(-\infty)=1 \]