A statistical test problem consists of a null hypothesis \(H_0\) and an alternative hypothesis \(H_1\) that are mutually exclusive and involve statements about the entire distribution or about specific parameters of a characteristic of interest in the population. A test statistic for a statistical test problem is a function \(T = g(X_1, \dots, X_n)\) of sample variables \(X_1, \dots, X_n\) with the following properties: - Sensitivity for the test problem: Based on the value \(t = g(x_1, \dots, x_n)\) resulting from the realizations \(x_1, \dots, x_n\) it can be judged whether \(H_0\) or \(H_1\) is more likely to be true for the population - Known distribution under the null hypothesis: The distribution of T under the assumption that the null hypothesis is true (test distribution) can be determined

Steps to test a hypotheses:

  1. Find \(H_0\) and \(H_1\)
  2. Find which values you have given and select a proper test
  3. Calculate the test size
  4. Compare your result with the rules of decision-making to find out, wether \(H_0\) is rejected or not

Approximate binomial test

For large samples. Can be used if the population is unknown.

(Gaussian) Z-test for the expected value of a normal distribution

\(o^2\) is known, but \(\mu\) unknown

T-test for the expected value of a normal distribution

\(o^2\) and \(\mu\) are unknown

[Errors in statistical tests] coming soon!

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