Simple Hypothesis and Composite Hypothesis

A simple hypothesis is one in which all parameters of the distribution are specified. For example, if the heights of college students are normally distributed with {\sigma ^2} = 4, the hypothesis that its mean \mu is, say,62'', that is {H_o}:\mu =  62, we have stated a simple hypothesis, as the mean and variance together specify a normal distribution completely. A simple hypothesis, in general, states that \theta = {\theta _o} where {\theta _o} is the specified value of a parameter \theta , (\theta may represent \mu ,p,{\mu _1} - {\mu _2} 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 {H_o}:\mu > 62 (and {\sigma ^2} = 4) or{H_o}:\mu =  62 and {\sigma ^2} < 4, the hypothesis becomes a composite hypothesis because we cannot know the exact distribution of the population in either case. Obviously, the parameters \mu >  62'' and{\sigma ^2} < 4 have more than one value and no specified values are being assigned. The general form of a composite hypothesis is \theta \leqslant {\theta _o} or \theta  \geqslant {\theta _o}, that is the parameter \theta does not exceed or does not fall short of a specified value {\theta _o}. The concept of simple and composite hypotheses applies to both 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 {H_o}:\mu =  62 or p > 0.5. A hypothesis is called an inexact hypothesis when it indicates more than one possible values for the parameter such as {H_o}:\mu \ne 62 or {H_o}:p = 62. A simple hypothesis must be an exact one while an exact hypothesis is not necessarily a simple hypothesis. An inexact hypothesis is a composite hypothesis.