# Critical value

In a test of hypothesis, a critical value is a number that separates two regions:

1. the critical region, that is, the set of values of the test statistic that lead to a rejection of the null hypothesis;

2. the acceptance region, that is, the set of values for which the null is not rejected.

## Notation

In what follows we are going to use the following notation:

• is the test statistic (e.g., a z-statistic);

• is the critical region;

• is the acceptance region.

Thus, the null hypothesis is rejected ifand it is not rejected if

## Definition

Here is a formal definition.

Definition A critical value is a boundary of the acceptance region .

Let us make an example.

Example If the acceptance region is the intervalthen the critical region isThe critical values are the two boundariesIf, for example, the value of the test statistic is , then belongs to the acceptance region and the null hypothesis is not rejected.

## One-tailed tests

A test is called one-tailed if there is only one critical value .

There are two cases:

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

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

The following table summarizes the two cases by using the symbols introduced above.

### How is the critical value determined in left-tailed tests?

Typically, 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 .

For a left-tailed test, we have

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,

Thus, we can writewhere 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 that solves the equation

We explain below how to solve it.

### How is the critical value determined in right-tailed tests?

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

Thus, we need to solve the equation

### How are the equations solved?

For the most common distributions such as the normal distribution and the t distribution, the equationsand have no analytical solution.

The reason is that the inverse of the cdf is not known in closed form.

So, for example, we cannot compute analytically the solution of the first equation as

However, virtually any calculator or statistical software has pre-built functions that allow us to easily solve these equations 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.

### Example

Let us make an example of a left-tailed test.

Suppose that the size of the test is , which means that we are happy with a 5% probability of incorrectly rejecting the null when it is true.

Suppose that our test statistic has a standard normal distribution.

Then, we need to findwhere is the cumulative distribution function of a standard normal distribution.

We are going to use a free web app called Wolfram Alpha to find .

Here is the result.

Thus, the critical value is .

We are going to reject the null hypothesis if is less than .

## Two-tailed tests

A test is called two-tailed if there are two critical values and and the null hypothesis is rejected only if

We assume without loss of generality that .

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

### 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:

By making again the assumption that the test statistic is a continuous random variable, we obtainwhere is the distribution function of .

### How is the equation solved?

Our problem is to solve one equation in two unknowns ( and ).

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.

Some possibilities are to:

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 has a normal or a Student's t distribution and both of these distributions are symmetric.

### Solution for symmetric distributions (practically relevant)

A distribution is symmetric (around zero) whenfor any number .

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

Without loss of generality, we can assumewith .

It follows that the size of the test can be written asand the equation to solve becomes

This is an equation in one unknown () 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.

## 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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