538 CHAPTER 10 Chi-Square Tests and the F-Distribution The Chi-Square Independence Test After finding the expected frequencies, you can test whether the variables are independent using a chi-square independence test. A chi-square independence test is used to test the independence of two variables. Using this test, you can determine whether the occurrence of one variable affects the probability of the occurrence of the other variable. DEFINITION Before performing a chi-square independence test, you must verify that (1) the observed frequencies were obtained from a random sample and (2) each expected frequency is at least 5. To perform a chi-square independence test, these conditions must be met. 1. The observed frequencies must be obtained using a random sample. 2. Each expected frequency must be greater than or equal to 5. If these conditions are met, then the sampling distribution for the test is approximated by a chi-square distribution with d.f. = 1r - 121c - 12 degrees of freedom, where r and c are the number of rows and columns, respectively, of a contingency table. The test statistic is x 2 = Σ1O - E22 E where O represents the observed frequencies and E represents the expected frequencies. The Chi-Square Independence Test To begin the independence test, you must first state a null hypothesis and an alternative hypothesis. For a chi-square independence test, the null and alternative hypotheses are always some variation of these statements. H0: The variables are independent. Ha: The variables are dependent. The expected frequencies are calculated on the assumption that the two variables are independent. If the variables are independent, then you can expect little difference between the observed frequencies and the expected frequencies. When the observed frequencies closely match the expected frequencies, the differences between O and E will be small and the chi-square test statistic will be close to 0. As such, the null hypothesis is unlikely to be rejected. For dependent variables, however, there will be large discrepancies between the observed frequencies and the expected frequencies. When the differences between O and E are large, the chi-square test statistic is also large. A large chi-square test statistic is evidence for rejecting the null hypothesis. So, the chi-square independence test is always a right-tailed test. Picturing the World A researcher wants to determine whether a relationship exists between where people work (workplace or home) and their educational attainment. The results of a random sample of 275 employed persons are shown in the contingency table. (Adapted from U.S. Bureau of Labor Statistics) Where they work Educational attainment Workplace Home Less than high school 16 2 High school diploma 56 10 Some college 49 11 BA degree or higher 87 44 Can the researcher use this sample to test for independence using a chi-square independence test? Why or why not?
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