Previously, we have learned about hypothesis testing for categorical data. These tests range from the [latex]\chi^2[/latex] goodness of fit test to the test of homogeneity to the test for independence. The one thing all of these tests have in common is that the variables of interest are categorical.

However, sometimes, our data set does not fit the [latex]\chi^2[/latex] test for independence, particularly when our expected counts are less than 5.  Instead, as long as we combine categories into a 2×2 contingency table, we can use a different test called Fisher’s Exact Test.

Let’s look at an example.

An independent researcher wants to determine a relationship between the color of a motorcyclist’s helmet and whether an injury was sustained in a crash. They randomly obtain a sample of data and organize that data into the following contingency table.

As stated above, sometimes, our data set does not satisfy the conditions for [latex]\chi^2[/latex] test for independence. But, by combining categories in a [latex]2 \times 2[/latex] contingency table, we can use a test called Fisher’s Exact Test of Independence. Fisher’s Exact Test is used for data in a [latex]2 \times 2[/latex] contingency table where one or more of the expected frequencies are less than five and certain conditions (detailed later) are met. We primarily use this test when the sample size is small. This test will provide us with an exact [latex]P[/latex]-value and does not require any approximations.