Two-Sided Gauß Test

A way to statistically infer if the Expectation from experiments does not equal the real Expectation. $H_0$ gets accepted when the Test Statistic attains values typical of $H_0$ (in this case values close to $\mu_0$ , small in modulus).

$H_0 : \mu=\mu_0$
$H_1 : \mu \neq \mu_0$

Acceptance Domain $I=\left[z_{\alpha / 2}, z_{1-\alpha / 2}\right]$ .

Test Decision (Accept if)

$t \in\left[z_{\alpha / 2}, z_{1-\alpha / 2}\right] \Longleftrightarrow \alpha \leq 2 \min \{\Phi(t), 1-\Phi(t)\}=2 \Phi(-|t|)$

Confidence Interval

$t \in\left[z_{\alpha / 2}, z_{1-\alpha / 2}\right] \Longleftrightarrow \mu_0 \in\left[\bar{x}-z_{1-\alpha / 2} \frac{\sigma}{\sqrt{n}}, \bar{x}+z_{1-\alpha / 2} \frac{\sigma}{\sqrt{n}}\right]$

Errors

Type 1 Error:

\mathbb{P}_{\mu_0}\left(\text { reject } H_0\right)=\mathbb{P}_{\mu_0}\left(T_n<z_{\alpha / 2}\right)+\mathbb{P}_{\mu_0}\left(T_n>z_{1-\alpha / 2}\right)=\alpha / 2+1-(1-\alpha / 2)=\alpha

Type 2 Error:

\begin{aligned} \beta(\mu) &=\mathbb{P}_\mu\left(\operatorname{accept} H_0\right)=\Phi_\lambda\left(z_{1-\alpha / 2}\right)-\Phi_\lambda\left(z_{\alpha / 2}\right) \\ &=\Phi_\lambda\left(z_{1-\alpha / 2}\right)+\Phi_{-\lambda}\left(z_{1-\alpha / 2}\right)-1 \quad\left(\mu \in H_1\right) \end{aligned}

Power Function

The Power Function is:

G(\mu)=1-\Phi_\lambda\left(z_{1-\alpha / 2}\right)+\Phi_\lambda\left(z_{\alpha / 2}\right)

Bildschirmfoto 2022-05-31 um 11.18.58.png

Critics for this Test

Detecting Inequality / Difference vs. Detecting Equality (impossible with infinite many values)
often $\mu\in[\mu_0-\delta,\mu_0+\delta]$ acceptable (less restrictive)
or equivalence test for $H_0$ : $|\mu-\mu_0|>\delta$ and for $H_1\leq\delta$

Confidence Interval

The Confidence Interval for the two-sided Gauß Test can be calculated like this:

Bildschirmfoto 2023-02-10 um 10.51.06.png