Gauß Test

Good approximation if sample size is large. Can be applied for lots of problems. Indirect Reasoning → Adopt point of other “person”

• One-Sided Gauß Test
  • Left-Sided Gauß Test
  • Right-Sided Gauß Test
• Two-Sided Gauß Test

Derivation of Test Statistic and Power Function

We know that the Arithmetic Mean from the Standard Estimators has Normal Distribution with Variance ${\sigma }^2/n$.

So we can now calculate the Standardized Mean (comes from subtracting the mean and dividing by the square root of the variance) looks like this: $T_n:=\frac{{\bar{X}}_{(n)}-{\mu }_0}{\sigma /\sqrt{n}}$ This now has Standard Normal Distribution with its corresponding CDF and p-quantiles.

We now want to create a new Random Variable which is the standardized $T_n$ plus the standardized distance between the two expectations from the hypothesis.

$Z=T_n+\frac{\mu -{\mu }_0}{\sigma }\sqrt{n}=T_n+\lambda$ $P_{\mu }(Z\le z)=P_{\mu }(T_n+\lambda \le z)$ We know the distribution of $T_n$ which is the standard normal distribution. We thus want to reformulate the equation to: $P_{\mu }(T_n\le z-\lambda )$ which we now know the distribution of. We can write $P_{\mu }(T_n\le z-\lambda )=\varphi (z-\lambda ):={\varphi }_{\lambda }(z)$ So basically a normal distribution which depends on the difference between the measured and real expectations.

${\varphi }_{\lambda }$ is strictly monotonically decreasing in $\mu$ which comes from the definition of $\lambda$. We can use this new definition of the Test Statistic $Z$ to define and draw the Power Function.

Other References

• https://de.wikipedia.org/wiki/Gauß-Test
• https://en.wikipedia.org/wiki/Student%27s_t-test → basically Gauß Test but for other non normal distributions or normal distributions with skewed variance
• https://en.wikipedia.org/wiki/Z-test