Simple Hypothesis and Composite Hypothesis

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 ${\sigma ^2} = 4$, and the hypothesis that its mean $\mu$ is, say, $62”$; that is, ${H_o}:\mu = 62$. 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 $\theta = {\theta _o}$ 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 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 ${H_o}:\mu = 62$ or $p > 0.5$. A hypothesis is called an inexact hypothesis when it indicates more than one possible value for the parameter, such as ${H_o}:\mu \ne 62$ or ${H_o}:p = 62$. A simple hypothesis must be exact while an exact hypothesis is not necessarily a simple hypothesis. An inexact hypothesis is a composite hypothesis.