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 22 0 α > . Large values of 2 0 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 np where 500 n = . This gives expected counts of 100, 50, 225 and 125 respectively. 6. Each expected count is i where 700 n = . This gives expected counts of 105, 210, 245 and 140 respectively. 7. (a) () 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 ii i O E OE OE E −− = (b) df = 4 – 1 = 3 (c) 2 0.05 7.815 = (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) () 22 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 ii i O E OE OE E χ −− = (b) df = 5 – 1 = 4 (c) 2 0.05 9.488 = (d) The test statistic is not in the (right-tailed) critical region so we do not reject 0 H . 9. (a) 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 O E = (b) df = 5 – 1 = 4 (c) 2 0.05 9.488 = (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 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 O E =
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Chapter 12 Inference on Categorical Data 622 (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 0.3 p = . 11. We summarize the observed and expected counts in the following table: () 22 2 0 Observed ( ) Expected / Brown 61 52 81 1.558 Yellow 64 56 64 1.143 Red 54 52 4 0.077 Blue 61 96 1225 12.760 Orange 96 80 256 3.2 Green 64 64 0 0 18.738 i i ii i O E OE OE E −− = Since all the expected cell counts are greater than or equal to 5, the requirements for the goodness-of-fit test are satisfied. Classical approach: The critical value, with df = 6 – 1 = 5, is 2 0.05 11.071 = . Since the test statistic is in the critical region, we reject the null hypothesis. P -value approach: Using the chi-square table, we find the row that corresponds to 5 degrees of freedom. The value of 18.738 is greater than 16.750, which has an area under the chi-square distribution of 0.005 to the right. Therefore, we have -value 0.005 P < . Since -value P α < , we reject the null hypothesis.
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This note was uploaded on 11/21/2011 for the course NGN 111 taught by Professor Ahmad during the Spring '11 term at American Dubai.

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

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