Hypothesis Goodness of Fit Test for Proportions of a Multinomial Population

1
Hypothesis (Goodness of Fit) Testfor Proportions
of a Multinomial Population
Multinomial Population a single population
consisting of more than one category.
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.
2
Hypothesis (Goodness of Fit) Testfor Proportions
of a Multinomial Population
4. Compute the value of the test statistic.
3
Example Finger Lakes Homes (A)

• Multinomial Distribution Goodness of Fit Test
• Finger Lakes Homes manufactures
• four models of prefabricated homes,
• a two-story colonial, a log cabin, a
• split-level, and an A-frame. To help
• in production planning, management
• would like to determine if previous
• customer purchases indicate that there is
• a preference in the style selected.

4
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
5
Example Finger Lakes Homes (A)

• Multinomial Distribution Goodness of Fit Test
• 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
6
Multinomial Distribution Goodness of Fit Test
7
Multinomial Distribution Goodness of Fit Test

• Rejection Rule

Reject H0 if c2 gt 7.815.
With ? .05 and k - 1 4 - 1 3
degrees of freedom
Do Not Reject H0
Reject H0
?2
7.815
8
Multinomial Distribution Goodness of Fit Test

• Conclusion

c2 10 gt 7.815
We reject, at the .05 level of
significance, the assumption that there is no
home style preference.
9
Now You Try. Pg. 467, 14
10
Test of Independence

• Uses the chi-square distribution to test for the
  independence of two variables

11
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.
12
Test of Independence Contingency Tables
4. Compute the test statistic.
13
Example Albers Brewery

• Contingency Table (Independence) Test
• Albers produces 3 types of beer light,
  regular, and dark. To help in the development of
  a new advertising campaign, the firm would like
  to know whether preferences for the 3 beers
  differ among male and female beer drinkers.
• A test of independence addresses the question
  of whether the beer preference (light, regular,
  or dark) is independent of the gender of the
  drinker (male, female).

14
Test of Independence Contingency Tables
Using sample data to test for the independence of
two variables
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.
15
Contingency Table (Independence) Test

• Hypotheses

H0 Beer preference is independent of gender Ha
Beer preference is not independent of gender

• Contingency Table

Beer Preference
16
Contingency Table (Independence) Test
A simple random sample of 150 is selected. After
tasting each beer, the individuals state their
preference. The sample results (observed
frequencies) are described below
17
Test of Independence Contingency Tables
Using sample data to test for the independence of
two variables
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.
18
Contingency Table (Independence) Test
19
Contingency Table (Independence) Test
Expected Frequencies
20
Test of Independence Contingency Tables
Using sample data to test for the independence of
two variables
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.
4. Compute the test statistic.
21
Computation of the ?2 Test Statistic
?2
22
Test of Independence Contingency Tables
Using sample data to test for the independence of
two variables
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.
4. Compute the test statistic.
23
Contingency Table (Independence) Test

• Rejection Rule
• Conclusion

Reject H0 if ?2 gt 5.99
Beer preference is NOT independent of the gender
of the drinker
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Now You Try. Page 475, 22
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End of Chapter 11