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Critical value

In a test of hypothesis, a test statistic is employed to decide whether to reject the null hypothesis or not. If the test statistic falls within a range of values called acceptance region, then the null hypothesis is not rejected. The bounds of the acceptance region, that is, the most extreme values for which the null is not rejected, are called critical values.

Definition

The following is a formal definition.

Definition Suppose $t_{n}$ is a test statistic such that the null hypothesis is rejected if[eq1]and it is not rejected if[eq2]where $C^{c}$ is the complement of $C$. If $C^{c}$ is an interval, then its extremes are called critical values.

Note that the set $C$ in the definition above is called the critical region of the test, while its complement $C^{c}$ is called the acceptance region.

Examples

The following is an example of a case where there are two critical values.

Example Suppose the critical region is[eq3]where $z$ is a real number. Then, the acceptance region is[eq4]Its extremes are $-z$ and $z$. Therefore $-z$ and $z$ are the critical values of the test.

In the next example there is only one critical value.

Example Suppose the critical region is[eq5]where $z$ is a real number. Then, the acceptance region is[eq6]Its only extreme is $z$. Therefore, $z$ is the critical value of the test.

One-tailed vs two-tailed tests

A statistical test in which there is only one critical value is often called a one-tailed test, while a test in which there are two critical values is called a two-tailed test.

Critical values and size of the test

The size of the test, that is, the probability of rejecting the null hypothesis when it is true, depends on how the critical values are set.

Typically, a statistician will set the critical values so as to achieve a desired size.

Example Suppose the critical region is[eq7]and the acceptance region is[eq8]Then the size of the test is[eq9]where [eq10] is the distribution function of the test statistic under the null hypothesis. If the statistician knows [eq11], then he can fix a value for $lpha $ and choose the critical value $z$ accordingly:[eq12]

More details

If you want to read a more detailed exposition of the concept of critical value and of related concepts, go to the lecture entitled Hypothesis testing.

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