Econ 3790: Business and Economics Statistics Instructor: Yogesh Uppal Email: yuppal@ysu.edu

Chapter 12 Goodness of Fit Test: A Multinomial Population Test of Independence

Hypothesis (Goodness of Fit) Test for Proportions of a Multinomial Population 1. State the null and alternative hypotheses. H0: The population follows a multinomial distribution with specified probabilities for each of the k categories Ha: The population does not follow a multinomial distribution with specified probabilities for each of the k categories Select a level of significance (a) and find a critical value from Chi-squared distribution with k-1 degrees of freedom.

Hypothesis (Goodness of Fit) Test for 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.

Hypothesis (Goodness of Fit) Test for Proportions of a Multinomial Population Select a random sample and record the observed frequency, fi , for each of the k categories. Assuming H0 is true, compute the expected frequency, ei , in each category by multiplying the category probability by the sample size.

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

Multinomial Distribution Goodness of Fit Test Example: Mosquito Lakes Homes Mosquito Lakes Homes manufactures four models of prefabricated homes, a two-story colonial, a log cabin, a split-level, and an A-frame. Check if previous customer purchases indicate that there is a preference in the style selected. The number of homes sold of each model for 100 sales over the past two years is shown below. Split- A- Model Colonial Log Level Frame # Sold 30 20 35 15

Multinomial Distribution Goodness of Fit Test Hypotheses H0: pC = pL = pS = pA = .25 Ha: The population proportions are not pC = .25, pL = .25, pS = .25, and pA = .25 where: pC = population proportion that purchase a colonial pL = population proportion that purchase a log cabin pS = population proportion that purchase a split-level pA = population proportion that purchase an A-frame

Test of Independence: Contingency Tables 1. Set up the null and alternative hypotheses. H0: The column variable is independent of the row variable Ha: The column variable is not independent of the row variable Select a level of significance (a) and find a critical value from Chi-squared distribution with (n-1)(m-1) degrees of freedom.

Test of Independence: Contingency Tables 3. Compute the test statistic. Select a random sample and record the observed frequency, fij , for each cell of the contingency table. Compute the expected frequency, eij , for each cell.

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

Contingency Table (Independence) Test Example: Mosquito Lakes Homes Each home sold by Mosquito Lakes Homes can be classified according to price and to style. Mosquito Lakes’ manager would like to determine if the price of the home and the style of the home are independent variables.

Contingency Table (Independence) Test Example: Mosquito Lakes Homes The number of homes sold for each model and price for the past two years is shown below. For convenience, the price of the home is listed as either $99,000 or less or more than $99,000. Price Colonial Log Split-Level A-Frame < $99,000 18 6 19 12 > $99,000 12 14 16 3