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Unformatted text preview: The Chi-Square Tests (Section 10.4) This section deals with independence/association between two categorical variables. We will cover three tests that are very similar in nature but differ in the conditions when they can be used. These are A) Goodness-of-tests B) Tests of homogeneity and C) Test of independence. Lets start with the easiest one. A) Goodness-of-fit Test (10.4.1 and 10.5.1) This is an extension of the one-population, one-sample, one-parameter problem where the random variable of interest is a categorical variable with 2 categories and the hypotheses were Ho: p = p o versus Ha: p p . We now extend the above test to the case of a categorical random variable with k (k 2) categories. Suppose we have a random variable that has k = 3 categories. Then the hypotheses of interest will be Ho: p 1 = p 10 , p 2 = p 20 , p 3 = p 30 vs. Ha: At least one of p i p i0 Where p i are the proportion of population units in the i th category and p i0 are the values of p i specified by the null hypothesis. To test these hypotheses we select a random sample of size n and count the number of sample units O bserved in each category (denoted by O i ). Next, we calculate the E xpected number of observations (E i ) in each category assuming Ho to be true, using E i = np i0 . Finally we compare the observed frequencies with the expected frequencies using the test statistic ( 29 2 2 2 ( ) 1 ~ k i i df i i O E E =- = , where df = k 1. Other steps of hypothesis testing are the same as before: 1) Assumptions a) Simple random samples from the population b) Categorical variable with k categories c) Large samples (E i 5 for all i) STA 6126 Chap 8, Page 1 of 22 2) Hypotheses Ho: p 1 = p 10 , p 2 = p 20 , p 3 = p 30 vs. Ha: At least one of p i p i0 3) Test Statistic: ( 29 2 2 2 ( ) 1 ~ k i i df i i O E E =- = , with df = (k1) 4) The p-value = ( 29 2 2 ( ) df cal P 5) Decision Same rule as ever, Reject Ho if the p-value . 6) Conclusion Same as before, explain the decision in simple English for the layman. Example: Suppose we suspect that a die (used in a Las Vegas Casino) is loaded. To see if this suspicion is warranted we roll the die 600 times and observe the frequencies given in Table 10.1 below. The hypotheses of interest are Ho: p 1 = p 2 = p 3 = p 4 = p 5 = p 6 = 1/6 vs. Ha: At least one of the p i 1/6. Lets test these hypotheses using the six steps of hypothesis testing. First we need to check if all of the conditions are satisfied: 1) Assumptions Satisfied? a) Simple random samples from the population Yes b) Categorical variable with k categories Yes, k = 6 c) Large samples (E i 5 for all i) Yes, look at Table 10.1 2) Hypotheses: Ho: p 1 = p 2 = p 3 = p 4 = p 5 = p 6 = 1/6 vs. Ha: At least one of the p i 1/6....
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