Mathematical Statistics

3Statistical Tests and Confidence Estimations

3.1 Basic Ideas of Test Theory

Sometimes the aim of investigation is neither to determine certain statistics (to estimate parameters) nor to select something, but to test or to examine carefully considered hypotheses (assumptions, suppositions) and often also wishful notions on the basis of practical material. Also in this case a mathematical model is established where the hypothesis can be formulated in the form of model parameters. We want to start with an example.

Table potatoes are examined beside other things, whether they are infected by so‐called brown foulness. Since the potato under examination is cut for that reason, it is impossible to examine the whole production. Hence, the examiner takes at random a certain number n of potatoes from the produced amount of potatoes and decides to award the rating ‘table potatoes’ if the number r of low‐quality potatoes is less or equal to a certain number c, and otherwise he declines to do so. (For example, we can suppose that a quantity of potatoes is classified as table potatoes, if the portion p of damaged or bad potatoes is smaller than or equal to 3%.) This is a typical statistical problem, because it concludes from a random sample (the n examined potatoes) to a population (the whole amount of potatoes of a certain producer in a certain year).

The above described situation is a bit more complicated than that for estimation and selection problems, because evidently two wrong decisions ...