Testing hypotheses using the t test

Abstract

There is one major problem with using the z test and the normal distribution to test even simple hypotheses. This problem arises, in the case of the test for the significance of a single mean, because we must assume that we know the standard deviation of the variable in the population. Specifically, we must know the population standard deviation in order to estimate the standard error of the sample mean. In practice, of course, we rarely know the values of any of the population parameters. We only know the values of the sample statistics. For many years, statisticians were content to estimate the population standard deviation of a variable using the sample standard deviation of that variable. However, they later discovered that this practice often led to incorrect inferences.