528 CHAPTER 10 Chi-Square Tests and the F -Distribution Before performing a chi-square goodness-of-fit test, you must verify that (1) the observed frequencies were obtained from a random sample and (2) each expected frequency is at least 5. Note that when the expected frequency of a category is less than 5, it may be possible to combine the category with another one to meet the second requirement. To perform a chi-square goodness-of-fit 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 k - 1 degrees of freedom, where k is the number of categories. The test statistic is x 2 = Σ1O - E22 E where O represents the observed frequency of each category and E represents the expected frequency of each category. The Chi-Square Goodness-of-Fit Test 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. However, when there are large discrepancies between the observed frequencies and the expected frequencies, the differences between O and E will be large, resulting in a large chi-square test statistic. A large chi-square test statistic is evidence for rejecting the null hypothesis. So, the chi-square goodness-of-fit test is always a right-tailed test. Performing a Chi-Square Goodness-of-Fit Test In Words In Symbols 1. Verify that the observed frequencies Ei = npi Ú 5 were obtained from a random sample and each expected frequency is at least 5. 2. Identify the claim. State the null and State H0 and Ha. alternative hypotheses. 3. Specify the level of significance. Identify a. 4. Identify the degrees of freedom. d.f. = k - 1 5. Determine the critical value. Use Table 6 in Appendix B. 6. Determine the rejection region. 7. Find the test statistic and sketch the x 2 = Σ1O - E22 E sampling distribution. 8. Make a decision to reject or fail to If x 2 is in the rejection reject the null hypothesis. region, then reject H0. Otherwise, fail to reject H0. 9. Interpret the decision in the context of the original claim. GUIDELINES Study Tip Remember that a chi-square distribution is positively skewed and its shape is determined by the degrees of freedom. Its graph is not symmetric, but it appears to become more symmetric as the degrees of freedom increase, as shown in Section 6.4.
RkJQdWJsaXNoZXIy NjM5ODQ=