1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Chapter 12 Tests of Goodness of Fit and Independence n Goodness of Fit Test: A Multinomial.

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1 1 Slide © 2008 Thomson South-Western. All Rights Reserved Chapter 12 Tests of Goodness of Fit and Independence n Goodness of Fit Test: A Multinomial Population Goodness of Fit Test: Poisson Goodness of Fit Test: Poisson and Normal Distributions and Normal Distributions Test of Independence Test of Independence

2 2 Slide © 2008 Thomson South-Western. All Rights Reserved Hypothesis (Goodness of Fit) Test for Proportions of a Multinomial Population 1. Set up the null and alternative hypotheses. 2. Select a random sample and record the observed frequency, f i, for each of the k categories. frequency, f i, for each of the k categories. 3. Assuming H 0 is true, compute the expected frequency, e i, in each category by multiplying the frequency, e i, in each category by multiplying the category probability by the sample size. category probability by the sample size.

3 3 Slide © 2008 Thomson South-Western. All Rights Reserved Hypothesis (Goodness of Fit) Test for Proportions of a Multinomial Population 4. Compute the value of the test statistic. Note: The test statistic has a chi-square distribution with k – 1 df provided that the expected frequencies are 5 or more for all categories. Note: The test statistic has a chi-square distribution with k – 1 df provided that the expected frequencies are 5 or more for all categories. f i = observed frequency for category i e i = expected frequency for category i k = number of categories where:

4 4 Slide © 2008 Thomson South-Western. All Rights Reserved Hypothesis (Goodness of Fit) Test for Proportions of a Multinomial Population where  is the significance level and there are k - 1 degrees of freedom p -value approach: Critical value approach: Reject H 0 if p -value <  5. Rejection rule: Reject H 0 if

5 5 Slide © 2008 Thomson South-Western. All Rights Reserved Test of Independence: Contingency Tables 1. Set up the null and alternative hypotheses. 2. Select a random sample and record the observed frequency, f ij, for each cell of the contingency table. frequency, f ij, for each cell of the contingency table. 3. Compute the expected frequency, e ij, for each cell.

6 6 Slide © 2008 Thomson South-Western. All Rights Reserved Test of Independence: Contingency Tables 5. Determine the rejection rule. Reject H 0 if p -value <  or. 4. Compute the test statistic. where  is the significance level and, with n rows and m columns, there are ( n - 1)( m - 1) degrees of freedom.