Slides Prepared by

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Slides Prepared by JOHN S. LOUCKS St. Edwards
University
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Chapter 11 Comparisons Involving Proportionsand
a Test of Independence

• Inference about the Difference between the
• Proportions of Two Populations
• Hypothesis Test for Proportions of a
• Multinomial Population
• Test of Independence Contingency Tables

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Inferences About the Difference between the
Proportions of Two Populations

• Sampling Distribution of
• Interval Estimation of p1 - p2
• Hypothesis Tests about p1 - p2

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Sampling Distribution of

• Expected Value
• Standard Deviation

where n1 size of sample taken from population
1 n2 size of sample taken from population 2
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Sampling Distribution of
The sample sizes are sufficiently large if all
of these conditions are met
n1p1 5
n1(1 - p1) 5
n2p2 5
n2(1 - p2) 5
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Sampling Distribution of
p1 p2
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Interval Estimation of p1 - p2

• Interval Estimate

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Example MRA

• MRA (Market Research Associates) is conducting
• research to evaluate the effectiveness of
• a clients new advertising campaign.
• Before the new campaign began, a
• telephone survey of 150 households
• in the test market area showed 60
• households aware of the clients
• product.
• The new campaign has been initiated with TV
  and
• newspaper advertisements running for three weeks.

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Example MRA
A survey conducted immediately after the
new campaign showed 120 of 250 households aware
of the clients product. Does the data support
the position that the advertising campaign has
provided an increased awareness of the clients
product?
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Point Estimator of the Difference Betweenthe
Proportions of Two Populations
p1 proportion of the population of households
aware of the product after the new
campaign p2 proportion of the population of
households aware of the product
before the new campaign
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Interval Estimate of p1 - p2Large-Sample Case
For ?? .05, z.025 1.96

• .08 1.96(.0510)
• .08 .10

Hence, the 95 confidence interval for the
difference in before and after awareness of the
product is -.02 to .18.
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Using Excel to Developan Interval Estimate of p1
p2

• Formula Worksheet

Note Rows 16-251 are not shown.
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Using Excel to Developan Interval Estimate of p1
p2

• Value Worksheet

Note Rows 16-251 are not shown.
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Hypothesis Tests about p1 - p2

• Hypotheses
• Test Statistic

H0 p1 - p2 0
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Hypothesis Tests about p1 - p2

• Point Estimator of where p1 p2

where
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Hypothesis Tests about p1 - p2

• Can we conclude, using a .05 level of
  significance, that the proportion of households
  aware of the clients product increased after the
  new advertising campaign?

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Hypothesis Tests about p1 - p2

• Hypotheses

H0 p1 - p2 0
p1 proportion of the population of households
aware of the product after the new
campaign p2 proportion of the population of
households aware of the product
before the new campaign
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Hypothesis Tests about p1 - p2

• Rejection Rule
• Test Statistic

Reject H0 if z 1.645
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Hypothesis Tests about p1 - p2

• Conclusion

z 1.56
Using a .05, we cannot conclude that the
proportion of households aware of the clients
product increased after the new campaign.
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Using Excel to Conducta Hypothesis Test about p1
p2

• Formula Worksheet

Note Rows 17-251 are not shown.
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Using Excel to Conducta Hypothesis Test about p1
p2

• Value Worksheet

Note Rows 17-251 are not shown.
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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.
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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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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

• 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
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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

• Expected Frequencies
• Test Statistic

• e1 .25(100) 25 e2 .25(100) 25
• e3 .25(100) 25 e4 .25(100) 25

1 1 4 4 10
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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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Using Excel to Conducta Goodness of Fit Test

• Worksheet (showing data)

Note Rows 13-101 are not shown.
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Using Excel to Conducta Goodness of Fit Test

• Formula Worksheet

Note Columns A-B and rows 13-101 are not shown.
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Using Excel to Conducta Goodness of Fit Test

• Value Worksheet

Note Columns A-B and rows 13-101 are not shown.
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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, fij , 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 Finger Lakes Homes (B)

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

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Example Finger Lakes Homes (B)

• Contingency Table (Independence) Test
• 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 19 12 99,000 12
14 16 3
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Contingency Table (Independence) Test

• Hypotheses

H0 Price of the home is independent of the
style of the home that is purchased Ha Price
of the home is not independent of the
style of the home that is purchased
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Contingency Table (Independence) Test

• Expected Frequencies

Price Colonial Log Split-Level
A-Frame Total 19 12 55 99K
12 14 16
3 45 Total 30
20 35 15
100
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Contingency Table (Independence) Test

• Rejection Rule
• Test Statistic

Reject H0 if ?2 7.81
.1364 2.2727 . . . 2.0833 9.1486
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Contingency Table (Independence) Test

• Conclusion

c2 9.15 7.81
We reject, at the .05 level of
significance, the assumption that the price of
the home is independent of the style of home that
is purchased.
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Using Excel to Conducta Test of Independence

• Worksheet (showing data entered)

Note Rows 11-101 are not shown.
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Using Excel to Conducta Test of Independence

• Worksheet (showing Pivot Table)

Note Columns A-C (sample data) are not shown.
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Using Excel to Conducta Test of Independence

• Formula Worksheet

Note Columns A-C (sample data) are not shown.
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Using Excel to Conducta Test of Independence

• Value Worksheet

Note Columns A-C (sample data) are not shown.
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End of Chapter 11