# 6 - BUAD 310: Applied Business Statistics October 6, 2010 1...

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Outline for Today Chi square goodness of fit test Chi square test for independence 2
Chi-Square Distribution 3 2 χ Like the t -distributions, the chi-square distribution is described by the degrees of freedom (df) Chi-square distribution is skewed to the right and takes only positive values (≥ 0)

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Some Chi-Square Distributions 4 x y 0 10 20 30 40 50 0.0 0.10 0.20 0.30 Degrees of Freedom=2 x 0 10 20 30 40 50 0.05 0.15 Degrees of Freedom=5 x 0 10 20 30 40 50 0.04 0.08 0.12 Degrees of Freedom=7 x 0 10 20 30 40 50 0.02 0.06 Degrees of Freedom=17 χ 2 distributions
Chi Square Test for Goodness of Fit Consider the outcome of a multinomial experiment where each of n randomly selected items is classified into one of k groups Let f i = number of items classified into group i ( i -th observed frequency ) E i = np i = expected number in i -th group if p i is probability of being in group i ( i -th expected frequency ) To check whether the observed frequencies are consistent with the assumed probabilities p 1 , …, p k , we should compare f i ’s to E i ’s 5

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Chi Square Goodness of Fit Test H 0 : probabilities are p 1 , p 2 , …, p k vs H a : the null hypothesis is not true Use the chi-square test statistic: Find the p-value using chi-square distribution. Reject H 0 at significance level α if p-value < α (p-value method) or (rejection point method). Note : Large values of the test statistic or small p-values provide evidence against H 0 . 6 2 2 1 ( ) k i i i i f E = E = - χ 2 2 α χ
P-Value After you have computed the test statistic use chi-square distribution with k -1 degrees of freedom (df) to find the p-value ( right tail probability ) 7

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An Example: Microwave Oven 8 Microwave oven wholesaler wishes to compare consumer preferences in Milwaukee with the historical market shares in Cleveland. If the consumer preferences in Milwaukee are substantially different, the wholesaler will consider changing its policies for stocking ovens. The wholesaler examines a random sample of 400 Milwaukee consumers. Cleveland Milwaukee Expected Brand Market Share Frequency Frequency 1 20% 102 80 2 35% 121 140 3 30% 120 120 4 15% 57 60
Chi Square Testing : Microwave Oven 9 H 0 : p 1 = 0.20, p 2 = 0.35, p 3 = 0.30, p 4 = 0.15 H a : H 0 fails to hold Cleveland Milwaukee Expected Brand Market Share Frequency

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