620 Chapter 12 Inference on Categorical Data12.1 Goodness of Fit Test 1. These procedures are for testing whether sample data are a good fit with a hypothesized distribution. 2.The 2χgoodness of fit tests are always right tailed because the numerator in the test statistic is squared, making every test statistic other than a perfect fit positive. So, we are measuring if 220αχχ>. Large values of 20χindicate that the sample data are far from their expected values and so lead one to reject the null hypothesis that the data fit a specified distribution. 3.The sample data must be obtained one by random sampling, all expected frequencies must be greater than or equal to 1, and no more than 20% of the expected frequencies should be less than 5. 4.If the expected count of a category is less than one, two or more categories can be combined together so that all expected values are at least one. Alternatively, the sample size can be increased to achieve the desired expected counts. 5.Each expected count is in p⋅where 500n=. This gives expected counts of 100, 50, 225 and 125 respectively. 6. Each expected count is in p⋅where 700n=. This gives expected counts of 105, 210, 245 and 140 respectively. 7. (a) ()()()2220Observed ()Expected /30252512025251282590.36222590.362.72iiiiiiiOEOEOEEχ−−=(b) df = 4 – 1 = 3 (c) 20.057.815χ=(d) The test statistic is not in the (right-tailed) critical region so we do not reject 0H.
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Section 12.1 Goodness of Fit Test 621 8. (a) ()()()2220Observed ()Expected /384040.14540250.625414010.0253340491.225434090.2252.200iiiiiiiOEOEOEEχ−−=(b) df = 5 – 1 = 4 (c) 20.059.488χ=(d) The test statistic is not in the (right-tailed) critical region so we do not reject 0H. 9. (a) ()()()2220Observed ()Expected /11.60.360.2253825.6153.766.006132153.6466.563.038440409.6924.162.256389409.6424.361.036312.561iiiiiiiOEOEOEEχ−−=(b) df = 5 – 1 = 4 (c) 20.059.488χ=(d) The test statistic is in the (right-tailed) critical region so we reject 0H. There is sufficient evidence to reject the claim that the random variable Xis binomial with 4n=and 0.8p=. 10. (a) ()()()2220Observed ()Expected /260240.1396.011.649400411.6134.560.327280264.6237.160.8965075.6655.368.669108.13.610.44611.987iiiiiiiOEOEOEEχ−−=