ABSTRACT
The key question in hypothesis testing is a proper choice of a test statistic that would minimize the probability of an erroneous decision. Unfortunately, it is impossible to minimize the probabilities of errors of both types (Type I and Type II): decreasing one of them comes at the expense of increasing the other. Ideally, one would like to find a test with minimal error probabilities of both types. A trivial test that never rejects the null hypothesis regardless of the data evidently has a zero probability of a Type I error but, on the other hand, the probability of a Type II error is one. For simple hypotheses one compares the performance of two different tests with the same significance level by their powers and looks for the maximal power test. In practical experiments, the data are analyzed sequentially as they are collected and further sampling is stopped once there is enough evidence for making a conclusion.
