GoodnessofFit Tests and Contingency Analysis

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Goodness-of-Fit Testsand Contingency Analysis

• Ch. 13

?2
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Hypothesis (Goodness of Fit) Testfor Proportions
of a Multinomial Population
Multinomial Population a single population
consisting of more than one category.
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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.

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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, oi , 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.
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Hypothesis (Goodness of Fit) Testfor Proportions
of a Multinomial Population
4. Compute the value of the test statistic.
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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.

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Multinomial Distribution Goodness of Fit Test

• Hypotheses

H0 ?C ?L ?S ?A .25 HA The population
proportions are not as stated above
where ?C population proportion that purchase
a colonial ?L population proportion that
purchase a log cabin ?S population proportion
that purchase a split-level ?A population
proportion that purchase an A-frame
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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
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Multinomial Distribution Goodness of Fit Test
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Multinomial Distribution Goodness of Fit Test

• Rejection Rule

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

• Conclusion

c2 10 7.815
We reject, at the .05 level of
significance, the assumption that there is no
home style preference.
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Now You Try

• In the brochure Colors made available by
  MM/MARS, maker of MM chocolate candies, the
  traditional distribution of colors for the plain
  candies is said to be as follows
• In a follow-up study, 1-lb. bags were used to
  determine whether the represented percentages
  were valid. The following results were obtained
  for a sample of 506 candies.

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Now You Try

• Use ? .05 to determine whether the observed
  data support the percentages reported by the
  company.

H0 ?Br .30, ?Y .20, ?R .20, ?O .10, ?G
.10, ?Bl .10 HA The population proportions
are not as stated above
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Now You Try
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Now You Try

• ?2.05 11.070 (5 degrees of freedom)
• 29.51 11.070 ? Reject H0
• The percentages reported by the company have
  changed.

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Test of Independence

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

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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).

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

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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, oij , for each cell of
the contingency table.
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Contingency Table (Independence) Test

• Hypotheses

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

• Contingency Table

Beer Preference
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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
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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, oij , for each cell of
the contingency table.
3. Compute the expected frequency, eij , for
each cell.
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Expected Frequencies
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Expected Frequencies
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Expected Frequencies
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Expected Frequencies
Expected Frequencies
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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, oij , for each cell of
the contingency table.
3. Compute the expected frequency, eij , for
each cell.
4. Compute the test statistic.
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Computation of the ?2 Test Statistic
?2
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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, oij , for each cell of
the contingency table.
3. Compute the expected frequency, eij , for
each cell.
4. Compute the test statistic.
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Contingency Table (Independence) Test

• Rejection Rule
• Conclusion

Reject H0 if ?2 5.99
Beer preference is NOT independent of the gender
of the drinker
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Now You Try

• In a study of brand loyalty in the automotive
  industry, new-car customers were asked whether
  the make of their new car was the same as the
  make of their previous car. The breakdown of 600
  responses shows the brand loyalty for domestic,
  European, and Asian cars.

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Now You Try

• Conduct a hypothesis test to determine whether
  brand loyalty is independent of the make of the
  car. Use ? .05. What is your conclusion?
• Observed Frequency (oij)

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Now You Try

• Expected Frequency (eij)

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Computation of the ?2 Test Statistic
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Now You Try

• Conclusion

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End of Chapter 13