A simple hypothesis is one in which all parameters of the distribution are specified. For example, the heights of college students are normally distributed with , and the hypothesis that its mean is, say, ; that is, . So we have stated a simple hypothesis, as the mean and variance together specify a normal distribution completely. A simple hypothesis, in general, states that where is the specified value of a parameter , ( may represent etc).
A hypothesis which is not simple (i.e. in which not all of the parameters are specified) is called a composite hypothesis. For instance, if we hypothesize that (and ) or and , the hypothesis becomes a composite hypothesis because we cannot know the exact distribution of the population in either case. Obviously, the parameters and have more than one value and no specified values are being assigned. The general form of a composite hypothesis is or ; that is, the parameter does not exceed or does not fall short of a specified value . The concept of simple and composite hypotheses applies to both the null hypothesis and alternative hypothesis.
Hypotheses may also be classified as exact and inexact. A hypothesis is said to be an exact hypothesis if it selects a unique value for the parameter, such as or . A hypothesis is called an inexact hypothesis when it indicates more than one possible value for the parameter, such as or . A simple hypothesis must be exact while an exact hypothesis is not necessarily a simple hypothesis. An inexact hypothesis is a composite hypothesis.