604 CHAPTER 11 Goodness-of-Fit and Contingency Tables 21.Equivalent Tests A x2 test involving a 2 * 2 table is equivalent to the test for the difference between two proportions, as described in Section 9-1. Using Table 11-1 on page 577 from the Chapter Problem, verify that the x2 test statistic and the z test statistic (found from the test of equality of two proportions) are related as follows: z2 = x2. Also show that the critical values have that same relationship. 22.Using Yates’s Correction for Continuity The chi-square distribution is continuous, whereas the test statistic used in this section is discrete. Some statisticians use Yates’s correction for continuity in cells with an expected frequency of less than 10 or in all cells of a contingency table with two rows and two columns. With Yates’s correction, we replace a1O - E22 E with a1 O - E - 0.522 E Given the contingency table in Exercise 9 “Four Quarters the Same as $1?” find the value of the x2 test statistic using Yates’s correction in all cells. What effect does Yates’s correction have? 11-2 Beyond the Basics Chapter Quick Quiz Exercises 1–5 refer to the sample data in the following table, which summarizes the frequencies of 500 digits randomly generated by Statdisk. Assume that we want to use a 0.05 significance level to test the claim that Statdisk generates the digits in a way that they are equally likely. Last Digit 0 1 2 3 4 5 6 7 8 9 Frequency 45 45 54 61 45 43 48 53 50 56 1. What are the null and alternative hypotheses corresponding to the stated claim? 2. When testing the claim in Exercise 1, what are the observed and expected frequencies for the first digit of 0? 3. Is the hypothesis test left-tailed, right-tailed, or two-tailed? 4. If using a 0.05 significance level to test the stated claim, find the number of degrees of freedom. 5. Given that the P-value for the hypothesis test is 0.720, what do you conclude? Does it appear that Statdisk generates the digits so that they are equally likely? Questions 6–10 refer to the sample data in the following table, which describes the fate of the passengers and crew aboard the Titanic when it sank on April 15, 1912. Assume that the data are a sample from a large population and we want to use a 0.05 significance level to test the claim that surviving is independent of whether the person is a man, woman, boy, or girl. Men Women Boys Girls Survived 332 318 29 27 Died 1360 104 35 18 6. Identify the null and alternative hypotheses corresponding to the stated claim. 7. What distribution is used to test the stated claim (normal, t, F, chi-square, uniform)? 8. Is the hypothesis test left-tailed, right-tailed, or two-tailed?
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