## 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 *x*2 goodness-of-fit test is and what it does.

•Calculate a *x*2 goodness-of-fit test.

•List the assumptions of the *x*2 goodness-of-fit test.

•Calculate the *x*2 test of independence.

•Interpret the *x*2 test of independence.

•Explain the assumptions of the *x*2 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 *x*2 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 *x*2 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 *χ*2goodness-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**

FREQUENCIES | PREGNANT | NOT PREGNANT |

Observed | 7 | 73 |

Expected | 14 | 66 |

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).

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