0673chapter11 - Chapter 11 Goodness-of-Fit and Contingency...

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Chapter 11 Goodness-of-Fit and Contingency Tables 11-2 Goodness-of-Fit 1. When digits are randomly generated, they should form a uniform distribution – i.e., a distribution in which each of the digits is equally likely. The test for goodness-of-fit is a test of the hypothesis that the sample data fit the uniform distribution. 2. The calculated χ 2 is a measure of the discrepancy between the hypothesized distribution and the sample data. An exceptionally large value of the χ 2 test statistic suggests a large discrepancy between the hypothesized distribution and the sample data – i.e., that there is not goodness-of–fit, and that the observed and expected frequencies are quite different. An exceptionally small value of the χ 2 test statistic suggests an extremely good fit – i.e., that the observed and expected values are almost identical. 3. O represents the observed frequencies. The twelve values for O are 5, 8, 7, 9, 13, 17, 11, 10, 10, 12, 8, 10. E represents the expected frequencies. The twelve values for E dependent on the expected (i.e., the hypothesized) distribution. If the hypothesized distribution is that weddings occur in different months with the same frequency, we expect the Σ O = 120 observations to be equally distributed among the 12 months so that each of the twelve values for E is 120/12 = 12. 4. The P-value indicates that the given observed frequencies have a 0.477 probability of occurring by chance if the population distribution is as hypothesized (i.e., if weddings do occur in the 12 months with equal frequency). Because this probability is high, the hypothesis cannot be rejected. There is not sufficient evidence to reject the claim that weddings occur in the 12 months with equal frequency. NOTE FOR EXERCISES 5 AND 6: The principle that the null hypothesis must contain the equality still applies. When the original claim is given in words, it will reduce either to a statement that the observed results fit or match (i.e., are equal to) some specified distribution or to a statement that the observed results are different from some specified distribution. In the first case the original claim is the null hypothesis, in the second case the original claim is the alternative hypothesis. 5. original claim: the actual outcomes agree with the expected frequencies H o : the actual outcomes agree with the expected frequencies H 1 : at least one outcome is not as expected α = 0.05 and df = 9 C.V. χ 2 = 2 χ α = 2 0.05 χ = 16.919 calculations: χ 2 = Σ [(O – E) 2 /E] = 8.185 P-value = χ 2 cdf (8.815,99,5) = 0.5156 conclusion: Do not reject Ho; there is not sufficient evidence to reject the claim that the actual outcomes agree with the expected frequencies. There is no reason to say the slot machine is not functioning as expected.
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Goodness-of-Fit SECTION 11-2 385 6. original claim: “A” students are not evenly distributed throughout the classroom H o : “A” students are evenly distributed throughout the classroom H 1 : “A” students are not evenly distributed throughout the classroom α = 0.05 and df = 2 C.V. χ 2 = 2 χ α = 2 0.05 χ = 5.991 calculations: χ 2 = Σ [(O – E) 2 /E] = 7.226 P-value = χ
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This note was uploaded on 03/07/2011 for the course STATISTICS 1022 taught by Professor Dr.kalluri during the Spring '11 term at University of South Florida - Tampa.

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0673chapter11 - Chapter 11 Goodness-of-Fit and Contingency...

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