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chapter12 - Chapter 12 Inference on Categorical Data 12.1...

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620 Chapter 12 Inference on Categorical Data 12.1 Goodness of Fit Test 1. These procedures are for testing whether sample data are a good fit with a hypothesized distribution. 2. The 2 χ goodness 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 2 2 0 α χ χ > . Large values of 2 0 χ indicate 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 i n p where 500 n = . This gives expected counts of 100, 50, 225 and 125 respectively. 6. Each expected count is i n p where 700 n = . This gives expected counts of 105, 210, 245 and 140 respectively. 7. (a) ( ) ( ) ( ) 2 2 2 0 Observed ( ) Expected / 30 25 25 1 20 25 25 1 28 25 9 0.36 22 25 9 0.36 2.72 i i i i i i i O E O E O E E χ = (b) df = 4 – 1 = 3 (c) 2 0.05 7.815 χ = (d) The test statistic is not in the (right-tailed) critical region so we do not reject 0 H .
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Section 12.1 Goodness of Fit Test 621 8. (a) ( ) ( ) ( ) 2 2 2 0 Observed ( ) Expected / 38 40 4 0.1 45 40 25 0.625 41 40 1 0.025 33 40 49 1.225 43 40 9 0.225 2.200 i i i i i i i O E O E O E E χ = (b) df = 5 – 1 = 4 (c) 2 0.05 9.488 χ = (d) The test statistic is not in the (right-tailed) critical region so we do not reject 0 H . 9. (a) ( ) ( ) ( ) 2 2 2 0 Observed ( ) Expected / 1 1.6 0.36 0.225 38 25.6 153.76 6.006 132 153.6 466.56 3.038 440 409.6 924.16 2.256 389 409.6 424.36 1.0363 12.561 i i i i i i i O E O E O E E χ = (b) df = 5 – 1 = 4 (c) 2 0.05 9.488 χ = (d) The test statistic is in the (right-tailed) critical region so we reject 0 H . There is sufficient evidence to reject the claim that the random variable X is binomial with 4 n = and 0.8 p = . 10. (a) ( ) ( ) ( ) 2 2 2 0 Observed ( ) Expected / 260 240.1 396.01 1.649 400 411.6 134.56 0.327 280 264.6 237.16 0.896 50 75.6 655.36 8.669 10 8.1 3.61 0.446 11.987 i i i i i i i O E O E O E E χ =
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