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\overset{\text{i.i.d.}}{\sim}\mathcal{N}(\mu,\sigma^2):\ I_\alpha = \Bigl[\bar{X}_n - z_{1-\alpha}\frac{\sigma}{\sqrt{n}}, \bar{X}_n + z_{1-\alpha}\frac{\sigma}{\sqrt{n}}\Bigr]$ +} + +{\scriptsize + \textbf{Beispiel}: Test mit Signifikanzniveau. + + Seien $X_1,\ldots,X_n \overset{\text{i.i.d.}}{\sim} \mathcal{N}(\theta, \sigma^2)$.\\ + Man teste $H_0: \theta = 100$, $H_A: \theta > 100$ auf Niveau $\alpha$ mit Stichprobe $\bar{x}_n$.\\ + $$ + \bar{X}_n - z_{1-\alpha}\frac{\sigma}{\sqrt{n}} = 100 \implies + \frac{\bar{X}_n - 100}{\frac{\sigma}{\sqrt{n}}} = z_{1-\alpha} + $$ + $$ + \boxed{H_0:\quad T = \frac{(\hat{X}_n - 100)}{\frac{\sigma}{\sqrt{n}}}} + $$ + Der rechtsseitige Test ist dann: $T(\bar{x}_n) > z_{1-\alpha}$ +} + +{\scriptsize + \textbf{Beispiel}: p-Wert mit $\alpha$-Quantilen auswerten + + Sei $T(\omega)=2$ die Realisierung eines $2$-seitigen Gauss-Test für $H_0$.\\ + Der $p$-Wert ist: $\P\bigl[|Z|\geq 2\bigr] = 2(1-\Phi(2))$ + \begin{align*} + \P[Z \in K] &= \P\Bigl[Z \in (-\infty, 2)\cup(2,+\infty)\Bigr] & (\text{def. Gauss-Test}) \\ + &= \P\bigl[|Z|\geq 2 \bigr] & (\text{Umschreiben}) \\ + &= 2(\P\bigl[Z \geq 2 \bigr]) & (\text{Symmetrie}) \\ + &= 2(1-\P\bigl[|Z| < 2 \bigr]) & (\P[A] = 1-\P[\bar{A}]) \\ + &= \boxed{2(1-\Phi(2))} & (\text{def. } \Phi) + \end{align*} + + + Wir nutzen für $\alpha=0.005, \alpha'=0.025$:\\ + $z_{0.975} = z_{1-0.025} =1.96, z_{0.995} = z_{1-0.005} = 2.58$ + $$ + z_{1-\alpha'} = z_{0.975} = 1.96 < 2 < 2.58 = z_{0.995} = z_{1-\alpha} + $$ + Also lehnen wir $H_0$ bei $\alpha$ ab ($\alpha \leq p$), und bei $\alpha'$ an. ($\alpha' > p$) + +} \ No newline at end of file