Right-Sided Gauß Test

Also known as the sellers perspective. This test is used to accept H0H_0 if the Test Statistic attains values typical for H0H_0 (in this case small values).

  • H0:μ<μ0H_0:\mu<\mu_0
  • H1:μ≥μ0H_1:\mu\geq\mu_0

Acceptance Domain I=(−∞,z1−α]I=(-\infty,z_{1-\alpha}].

Test Decision (Accept if)

t∈(−∞,z1−α]⟺α≤1−Φ(t)t \in\left(-\infty, z_{1-\alpha}\right] \Longleftrightarrow \alpha \leq 1-\Phi(t)

Confidence Interval

t∈(−∞,z1−α]⟺μ0≥xˉ−z1−ασnt \in\left(-\infty, z_1-\alpha\right] \Longleftrightarrow \mu_0 \geq \bar{x}-z_{1-\alpha} \frac{\sigma}{\sqrt{n}}

Errors

The Type 1 Error is also called "embarrassing error" which we want to minimize:

Pμ( reject H0)=Pμ(Tn>z1−α)=1−Φλ(z1−α)<1−Φ0(z1−α)=1−(1−α)=α(μ∈H0)\begin{aligned} \mathbb{P}_{\mu}\left(\text { reject } H_{0}\right) &=\mathbb{P}_{\mu}\left(T_{n}>z_{1-\alpha}\right)=1-\Phi_{\lambda}\left(z_{1-\alpha}\right) \\ &<1-\Phi_{0}\left(z_{1-\alpha}\right)=1-(1-\alpha)=\alpha \quad\left(\mu \in H_{0}\right) \end{aligned}

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

β(μ)=Pμ(accept⁡H0)=Φλ(z1−α)(μ∈H1)\beta(\mu)=\mathbb{P}_\mu\left(\operatorname{accept} H_0\right)=\Phi_\lambda\left(z_{1-\alpha}\right) \quad\left(\mu \in H_1\right)

Both errors depend on the μ\mu that has been chosen or found from an experiment.

Power Function

The Power Function is:

G(μ)=1−Φλ(z1−α)=Φ−λ(zα)G(\mu)=1-\Phi_\lambda\left(z_{1-\alpha}\right)=\Phi_{-\lambda}\left(z_\alpha\right)

Bildschirmfoto 2022-05-31 um 10.54.26.png

Similar to the Left-Sided Gauß Test important statements can be generated from the power function curve.