596 CHAPTER 11 Goodness-of-Fit and Contingency Tables Procedure In conducting a test of homogeneity, we can use the same notation, requirements, test statistic, critical value, and procedures given in the Key Elements box from Part 1 on page 591 of this section, with this exception: Instead of testing the null hypothesis of independence between the row and column variables, we test the null hypothesis that the different populations have the same proportion of some characteristic. TABLE 11-7 Lost Wallet Experiment City A B C D E F G H I J K L M N O P Wallet Returned 85711586731424649 Wallet Not Returned 475174659118108683 The Lost Wallet Experiment EXAMPLE 3 Table 11-7 lists results from a Reader’s Digest experiment in which 12 wallets were intentionally lost in each of 16 different cities, including New York City, London, Amsterdam, and so on. Use a 0.05 significance level with the data from Table 11-7 to test the null hypothesis that the cities have the same proportion of returned wallets. The Reader’s Digest headline “Most Honest Cities: The Reader’s Digest Lost Wallet Test” implies that whether a wallet is returned is dependent on the city in which it was lost. Test the claim that the proportion of returned wallets is not the same in the 16 different cities. SOLUTION REQUIREMENT CHECK (1) Based on the description of the study, we will treat the subjects as being randomly selected and randomly assigned to the different cities. (2) The results are expressed as frequency counts in Table 11-7. (3) The expected frequencies are all at least 5. (All expected values are either 5.625 or 6.375.) The requirements are satisfied. The null hypothesis and alternative hypothesis are as follows: H0: Whether a lost wallet is returned is independent of the city in which it was lost. H1: A lost wallet being returned depends on the city in which it was lost. The accompanying StatCrunch display shows the test statistic of x2 = 35.388 (rounded) and the P-value of 0.002 (rounded). Because the P-value of 0.002 is less than the significance level of 0.05, we reject the null hypothesis of independence between the two variables. (“If the P is low, the null must go.”) INTERPRETATION We reject the null hypothesis of independence, so it appears that the proportion of returned wallets depends on the city in which they were lost. There is sufficient evidence to conclude that the proportion of returned wallets is not the same in the 16 different cities. Fisher’s Exact Test The procedures for testing hypotheses with contingency tables have the requirement that every cell must have an expected frequency of at least 5. This requirement is necessary for the x2 distribution to be a suitable approximation to the exact distribution of the x2 test statistic. Fisher’s exact test is often used for a 2 * 2 contingency table StatCrunch P q b th th Safest Seats in a Commercial Jet A study by aviation writer and researcher David Noland showed that sitting farther back in a commercial jet will increase your chances of surviving in the event of a crash. The study suggests that the chance of surviving is not the same for each seat, so a goodness-of-fit test would lead to rejection of the null hypothesis that every seat has the same probability of a passenger surviving. Records from the 20 commercial jet crashes that occurred since 1971 were analyzed. It was found that if you sit in business or first class, you have a 49% chance of surviving a crash; if you sit in coach over the wing or ahead of the wing, you have a 56% chance of surviving; and if you sit in the back behind the wing, you have a 69% chance of surviving. In commenting on this study, David Noland stated that he does not seek a rear seat when he flies. He says that because the chance of a crash is so small, he doesn’t worry about where he sits, but he prefers a window seat. A a a D s s b
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