Gauß Test

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

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\leq z)=P_{\mu}(T_n+\lambda\leq 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\leq z-\lambda)

which we now know the distribution of. We can write

P_{\mu}(T_n\leq z-\lambda)=\phi(z-\lambda):=\phi_{\lambda}(z)

So basically a normal distribution which depends on the difference between the measured and real expectations.

$\phi_{\lambda}$ 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