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011+Categorical+data

Thus we must to test the hypothesis h0 of

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Unformatted text preview: p1 is the total (marginal) probability to buy small car, etc. We would like to know (with α = 0.05) is their any dependence between size and manufacturer? Thus, we must to test the hypothesis H0 of independence. If the hypothesis of independence is true, then Since probabilities in last Table are unknown, for the test statistic we may estimate them by ratios p1 = ˆ n1 413 = = 0.413, n 1, 000 pA = ˆ nA 341 = = 0.341 etc. n 1, 000 Now E (x11 ) = np1 · pA = ˆˆ 413 · 341 n1 nA = = 140.83 n 1, 000 We obtain the following table 5 etc. A. Zhensykbaev Categorical data analysis Manufacturer A B C D Total Small 157 (140.83) 65 (79.3) 181 (158.2) 10 (34.7) 413 Intermediate 126 (135) 82 (76) 142 (151.7) 46 (33.3) 396 Large 58 (65.1) 45 (36.7) 60 (73.1) 28 (16) 191 Total 341 192 383 84 1,000 Test statistic is 3 4 2 χ= i=1 j =1 (nij − E (xij ))2 = 45.81. E (xij ) Rejected region is RR = {χ2 > χ2 = χ2.05 = 12.59}, α 0 where χ2.05 is chosen using the Table VII for the degrees of freedom df = 0 (r − 1)(c − 1) = (3 − 1)(4 − 1) = 6. Since 45.81> 12.59 we reject H0 . 6...
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