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).

• $H_0:\mu ={\mu }_0$
• $H_1:\mu \ne {\mu }_0$

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

Test Decision (Accept if)

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

Confidence Interval

$t\in \left[z_{\alpha /2},z_{1-\alpha /2}\right]⟺{\mu }_0\in \left[\bar{x}-z_{1-\alpha /2}\frac{\sigma }{\sqrt{n}},\bar{x}+z_{1-\alpha /2}\frac{\sigma }{\sqrt{n}}\right]$

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:

$$\begin{array}{rl}\beta (\mu )& ={\mathbb{P}}_{\mu }\left(\mathrm{accept}{H}_0\right)={\Phi }_{\lambda }\left(z_{1-\alpha /2}\right)-{\Phi }_{\lambda }\left(z_{\alpha /2}\right)\\ & ={\Phi }_{\lambda }\left(z_{1-\alpha /2}\right)+{\Phi }_{-\lambda }\left(z_{1-\alpha /2}\right)-1\left(\mu \in H_1\right)\end{array}$$

Power Function

The Power Function is: $G(\mu )=1-{\Phi }_{\lambda }\left(z_{1-\alpha /2}\right)+{\Phi }_{\lambda }\left(z_{\alpha /2}\right)$

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 $H_1\le \delta$

Confidence Interval

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