# Student test -test

A significance test for the mean value of a normal distribution.

## The single-sample Student test.

Let the independent random variables be subject to the normal law , the parameters and of which are unknown, and let a simple hypothesis : be tested against the composite alternative : . In solving this problem, a Student test is used, based on the statistic where are estimators of the parameters and , calculated with respect to the sample . When is correct, the statistic is subject to the Student distribution with degrees of freedom, i.e. where is the Student distribution function with degrees of freedom. According to the single-sample Student test with significance level , , the hypothesis must be accepted if where is the quantile of level of the Student distribution with degrees of freedom, i.e. is the solution of the equation . On the other hand, if then, according to the Student test of level , the tested hypothesis : has to be rejected, and the alternative hypothesis : has to be accepted.

## The two-sample Student test.

Let and be mutually independent normally-distributed random variables with the same unknown variance , and let  where the parameters and are also unknown (it is often said that there are two independent normal samples). Moreover, let the hypothesis : be tested against the alternative : . In this instance, both hypotheses are composite. Using the observations and it is possible to calculate the estimators for the unknown mathematical expectations and , as well as the estimators for the unknown variance . Moreover, let Then, when is correct, the statistic is subject to the Student distribution with degrees of freedom. This fact forms the basis of the two-sample Student test for testing against . According to the two-sample Student test of level , , the hypothesis is accepted if where is the quantile of level of the Student distribution with degrees of freedom. If then, according to the Student test of level , the hypothesis is rejected in favour of .