Chapter 12 Tests of Goodness of Fit and Independence

1. Chapter 12 Tests of Goodness of Fit and Independence • Goodness of Fit Test: A Multinomial Population • Test of Independence • Goodness of Fit Test: Poisson and Normal Distributions

2. Hypothesis (Goodness of Fit) Testfor Proportions of a Multinomial Population 1. Set up the null and alternative hypotheses. 2. Select a random sample and record the observed frequency, fi , for each of the k categories. 3. Assuming H0 is true, compute the expected frequency, ei , in each category by multiplying the category probability by the sample size.

3. Hypothesis (Goodness of Fit) Testfor Proportions of a Multinomial Population 4. Compute the value of the test statistic. where: fi = observed frequency for category i ei = expected frequency for category i k = number of 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.

4. Reject H0 if Hypothesis (Goodness of Fit) Testfor Proportions of a Multinomial Population 5. Rejection rule: Reject H0 if p-value <a p-value approach: Critical value approach: where  is the significance level and there are k - 1 degrees of freedom

5. Test of Independence: Contingency Tables 1. Set up the null and alternative hypotheses. 2. Select a random sample and record the observed frequency, fij , for each cell of the contingency table. 3. Compute the expected frequency, eij , for each cell.

6. Reject H0 if p -value <a or . Test of Independence: Contingency Tables 4. Compute the test statistic. 5. Determine the rejection rule. where  is the significance level and, with n rows and m columns, there are (n - 1)(m - 1) degrees of freedom.