1. Econ 3790: Business and Economics Statistics
Instructor: Yogesh Uppal

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

3. 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.

4. 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.

5. 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.

6. 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

7. 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

8. 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

9. 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.

10. 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.

11. 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.

12. 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.

13. 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,
> $99,