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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 boundaries of the acceptance region, that is, the most extreme values for which the null is not rejected, are called critical values.

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Definition

The following is a formal definition.

Definition Let $C $be a subset of the set of real numbers. Suppose $t_{n}$ is a test statistic such that the null hypothesis is rejected if[eq1] and 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.

One-tailed tests

A test is called one-tailed if there is only one critical value $z$. In particular, there are two cases:

  1. left-tailed test: the null is rejected only if[eq3]

  2. right-tailed test: the null is rejected only if [eq4]

The following table summarizes the two cases by using the symbols introduced in the definition above.[eq5]

How is the critical value determined?

Typically, $z$ is chosen so as to a achieve a desired size of the test.

Remember that the size is the probability of rejecting the null hypothesis when it is true. Denote it by $lpha $. For a left-tailed test, we have[eq6]In the majority of practical cases, the test statistic is a continuous random variable. As a consequence, the probability that it takes any specific value is equal to zero. In particular, [eq7]Thus, we can write[eq8]where $Fleft( z
ight) $ is the cumulative distribution function (cdf) of the test statistic.

All we need to do in order to determine the critical value is to find a $z$ that solves the equation[eq9]For the most common distributions such as the normal distribution and the t distribution, the equation has no analytical solution because the inverse of the cdf[eq10]is not known in closed form. However, virtually any calculator or statistical software has pre-built functions that allow to easily solve the equation numerically. The (old-fashioned) alternative is to look up the critical value in special tables called statistical tables. See this lecture if you want to know more about these alternatives.

Things are similar for right-tailed tests. In these tests, we have[eq11]

Thus, we have to solve the equation[eq12]which is solved exactly as in the left-tailed case (the only difference is that we need to replace $lpha $ with $1-lpha $).

Two-tailed tests

A test is called two-tailed if there are two critical values $z_{1}$ and $z_{2}$ and the null hypothesis is rejected only if[eq13]

We assume without loss of generality that $z_{1}<z_{2}$.

Thus, we can add a new line to the table shown in the previous section:[eq14]

How do you find the two critical values?

As in the case of a one-tailed test (see above), also in the two-tailed case the critical values are chosen so as to achieve a pre-defined size of the test.

The size can be computed as follows:[eq15]

By making again the assumption that the test statistic is a continuous random variable, we obtain[eq16]where $Fleft( z
ight) $ is the distribution function of $t_{n}$.

Our problem is to solve one equation in two unknowns ($z_{1}$ and $z_{2}$). There are potentially infinite solutions to the problem because one can choose one of the two critical values at will and choose the remaining one so as to solve the equation. There is no general rule for choosing one specific solution. A possibility is to try and find the solution which maximizes the power of the test in correspondence of a given alternative hypothesis. Another possibility is to find the solution which maximizes the length of the acceptance interval [eq17]. We do not discuss these possibilities here, but we refer the reader to Berger and Casella (2002). We instead discuss the case in which the test statistic has a symmetric distribution. This is the most relevant case in practice because in many tests $t_{n}$ has a normal or a Student's t distribution and both of these distributions are symmetric.

A distribution is symmetric (around zero) when[eq18]for any number $z$.

We can exploit this fact by making the additional assumption that the two critical values are opposite:[eq19]

Without loss of generality, we can assume[eq20]with $zgeq 0$.

It follows that the size of the test can be written as[eq21]and the equation to solve becomes[eq22]

This is an equation in one unknown ($z$) that can be solved using the methods (numeric inversion, tables, etc.) discussed in the previous section on one-tailed tests.

Summary

Everything we have said thus far is summarized by the following table.[eq23]

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.

References

Berger, R. L. and G. Casella (2002) "Statistical inference", Duxbury Advanced Series.

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