Left-Sided Gauß Test

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

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

Acceptance Domain $I=[z_{\alpha },\infty )$

Test Decision (Accept if)

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

Confidence Interval

$t\in \left[z_{\alpha },\infty \right)⟺{\mu }_0\le \bar{x}+z_1-\alpha \frac{\sigma }{\sqrt{n}}$

Errors

The Type 1 Error is also called “embarrassing error” which we want to minimize: ${\mathbb{P}}_{\mu }\left(\text{ reject }{H}_0\right)={\mathbb{P}}_{\mu }\left(T_n<z_{\alpha }\right)={\Phi }_{\lambda }\left(z_{\alpha }\right)\le {\Phi }_0\left(z_{\alpha }\right)=\alpha \left(\mu \in H_0\right)$

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

Power Function

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

One can see that large values for $\mu$ correspond to small values of the Power Function and thus correspond to small probability for Type 1 Error.

On the other hand small $\mu$ corresponds to high power and therefor small probability for Type 2 Error.

Also you can tell that the power is quite low when we get near ${\mu }_0$ on the $H_1$ side of the graph. This corresponds with a hight Type 2 Error.

You can try to get it to look like a 1-0 jump by increasing the sample size. Power will get arbitrarily large (also see Planning Test Size)

This test can also be done from the sellers perspective → Right-Sided Gauß Test.