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Ch 15 - Chi-Squared _Categorical Data_


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Unformatted text preview: in the table, so 6 terms to add up in the test statistic. The larger the test statistic, the “bigger” the differences between what we observed and what we would expect to see if H0 were true. So the larger the test statistic, the more evidence we have against the null hypothesis. Is a value of X 2 1.73 large enough to reject H0? We need to find the p‐value, the probability of getting an X 2 test statistic value as large or larger than the one we observed, assuming H0 is true. To do this we need to know the distribution of the X 2 test statistic under the null hypothesis. If H0is true, then X 2 has the 2 distribution with degrees of freedom = (r – 1)(c – 1) Brief motivation for the degrees of freedom formula: If you knew that 50% were boys you would know there were 50% girls (c – 1) If you know say 70% liked choc or van you would know 30% liked straw (r – 1) Find the p‐value for our ice cream example: Observed X 2 test statistic value = 1.73 df = (3 – 1)(2 – 1) = 2 Decision at a 5% significance level: (circle one) Reject H0 Fail to reject H0 Make a sketch and find the bounds for the p‐value … 0.25 < p‐value < 0.50. Conclusion: It appears that .... The distribution of ice cream preference is the same for the populations of boys and girls represented by these samples. 213 Test of Homogeneity Summary (Comparison of Several Populations) Assume: We have C independent random samples of size n1 , n2 ,..., nc from C populations. We measure 1 discrete response X that has r possible outcomes. Test: H0: The distribution for the response variable X is the same for all populations. Test Statistic: X 2 observed - expected2 expected where expected (row total)(column total) Total n If H0 is true, then X 2 has a 2 distribution with ( r 1)(c 1) degrees of freedom. The necessary conditions are: at least 80% of the expected counts are greater than 5 and none are less than 1. Try It! What is your Decision? For a chi‐square test of homogeneity, there are 3 populations and 4 possible values of the discrete characteristic. If H0 is true, that...
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