The Chi-Square ( x 2) Goodness-of-Fit test

The Chi-Square ( x 2) Goodness-of-Fit test

The Chi-Square ( x 2) Goodness-of-Fit test

Reference/Module

Learning Objectives

•Explain what the x2 goodness-of-fit test is and what it does.

•Calculate a x2 goodness-of-fit test.

•List the assumptions of the x2 goodness-of-fit test.

•Calculate the x2 test of independence.

•Interpret the x2 test of independence.

•Explain the assumptions of the x2 test of independence.

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The Chi-Square ( x 2) Goodness-of-Fit test: What It Is and What It Does

The  chi-square ( x 2) goodness-of-fit test  is used for comparing categorical information against what we would expect based on previous knowledge. As such, it tests what are called  observed frequencies  (the frequency with which participants fall into a category) against  expected frequencies  (the frequency expected in a category if the sample data represent the population). It is a nondirectional test, meaning that the alternative hypothesis is neither one-tailed nor two-tailed. The alternative hypothesis for a x2 goodness-of-fit test is that the observed data do not fit the expected frequencies for the population, and the null hypothesis is that they do fit the expected frequencies for the population. There is no conventional way to write these hypotheses in symbols, as we have done with the previous statistical tests. To illustrate the x2 goodness-of-fit test, let’s look at a situation in which its use would be appropriate.

chi-square (x2) goodness-of-fit test A nonparametric inferential procedure that determines how well an observed frequency distribution fits an expected distribution.

observed frequencies The frequency with which participants fall into a category.

expected frequencies The frequency expected in a category if the sample data represent the population.

Calculations for the  x 2 Goodness-of-Fit Test

Suppose that a researcher is interested in determining whether the teenage pregnancy rate at a particular high school is different from the rate statewide. Assume that the rate statewide is 17%. A random sample of 80 female students is selected from the target high school. Seven of the students are either pregnant now or have been pregnant previously. The χ2goodness-of-fit test measures the observed frequencies against the expected frequencies. The observed and expected frequencies are presented in  Table 21.1 .

TABLE 21.1 Observed and expected frequencies for χ2 goodness-of-fit example

As can be seen in the table, the observed frequencies represent the number of high school females in the sample of 80 who were pregnant versus not pregnant. The expected frequencies represent what we would expect based on chance, given what is known about the population. In this case, we would expect 17% of the females to be pregnant because this is the rate statewide. If we take 17% of 80 (.17 × 80 = 14), we would expect 14 of the students to be pregnant. By the same token, we would expect 83% of the students (.83 × 80 = 66) to be not pregnant. If the calculated expected frequencies are correct, when summed they should equal the sample size (14 + 66 = 80).