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).

$H_0:\mu\geq\mu_0$
$H_1:\mu<\mu_0$

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

Test Decision (Accept if)

$t \in\left[z_\alpha, \infty\right) \Longleftrightarrow \alpha \leq \Phi(t)$

Confidence Interval

$t \in\left[z_\alpha, \infty\right) \Longleftrightarrow \mu_0 \leq \bar{x}+z_1-\alpha \frac{\sigma}{\sqrt{n}}$

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) \leq \Phi_{0}\left(z_{\alpha}\right)=\alpha \quad\left(\mu \in H_{0}\right)

The Type 2 Error can get large as it would be ok:

\beta(\mu)=\mathbb{P}_\mu\left(\operatorname{accept} H_0\right)=1-\Phi_\lambda\left(z_\alpha\right) \quad\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)

Bildschirmfoto 2022-05-31 um 10.51.29.png

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.