Right-Sided Gauß Test

Also known as the sellers perspective. This test is used to accept $H_0$ if the Test Statistic attains values typical for $H_0$ (in this case small values).

• $H_0:\mu <{\mu }_0$
• $H_1:\mu \ge {\mu }_0$

Acceptance Domain $I=(-\infty ,z_{1-\alpha }]$.

Test Decision (Accept if)

$t\in \left(-\infty ,z_{1-\alpha }\right]⟺\alpha \le 1-\Phi (t)$

Confidence Interval

$t\in \left(-\infty ,z_1-\alpha \right]⟺{\mu }_0\ge \bar{x}-z_{1-\alpha }\frac{\sigma }{\sqrt{n}}$

Errors

The Type 1 Error is also called “embarrassing error” which we want to minimize:

$$\begin{array}{rl}{\mathbb{P}}_{\mu }\left(\text{ reject }{H}_0\right)& ={\mathbb{P}}_{\mu }\left(T_n>z_{1-\alpha }\right)=1-{\Phi }_{\lambda }\left(z_{1-\alpha }\right)\\ & <1-{\Phi }_0\left(z_{1-\alpha }\right)=1-(1-\alpha )=\alpha \left(\mu \in H_0\right)\end{array}$$

The Type 2 Error can get large as it would be ok: $\beta (\mu )={\mathbb{P}}_{\mu }\left(\mathrm{accept}{H}_0\right)={\Phi }_{\lambda }\left(z_{1-\alpha }\right)\left(\mu \in H_1\right)$ Both errors depend on the $\mu$ that has been chosen or found from an experiment.

Power Function

The Power Function is: $G(\mu )=1-{\Phi }_{\lambda }\left(z_{1-\alpha }\right)={\Phi }_{-\lambda }\left(z_{\alpha }\right)$

Similar to the Left-Sided Gauß Test important statements can be generated from the power function curve.