Chapter 12 Test of Independence

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Chapter 12 Test of Independence

• Test of Independence

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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.
5. Determine the rejection rule.
where ? is the significance level and, with n
rows and m columns, there are (n - 1)(m - 1)
degrees of freedom.
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Contingency Table (Independence) Test

• Example Finger Lakes Homes

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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Contingency Table (Independence) Test

• Example Finger 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
12
99,000 12 14 16
3
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Contingency Table (Independence) Test
Price Colonial Log Split-Level
A-Frame Total 99K Total
18 6 19
12 55
12 14 16
3 45
30 20 35
15 100
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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 if Price of the home is
  independent of the style of the home that is
  purchased

Price Colonial Log Split-Level
A-Frame Total 99K Total
55(30/100) 55(20/100) 55(.35) 55(.15)
45(.30) 45(.20) 45(.35) 45(.15)
30 20 35
15 100
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Contingency Table (Independence) Test

• Expected Frequencies if Price of the home is
  independent of the style of the home that is
  purchased

Price Colonial Log Split-Level
A-Frame Total 99K Total
16.5 11 19 .25 8.25
55
13.5 9 15.75 6.75
45
30 20 35
15 100
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Contingency Table (Independence) Test

• Rejection Rule

Reject H0 if p-value 7.815

• Test Statistic

.1364 2.2727 . . . 2.0833 9.134
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Contingency Table (Independence) Test

• Conclusion Using the p-Value Approach

Area in Upper Tail .10 .05 .025
.01 .005
c2 Value (df 3) 6.251 7.815 9.348
11.345 12.838
Because c2 9.134 is between 7.815 and 9.348,
the area in the upper tail of the distribution
is between .05 and .025.
The p-value hypothesis.
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Contingency Table (Independence) Test

• Conclusion Using the Critical Value Approach

c2 9.134 7.815
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
EXCEL
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End Chapter 12