Contingency Tables: Independence and Homogeneity

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Section 11-3

• Contingency Tables Independence and Homogeneity

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• A contingency table/two-way frequency table is a
  table in which frequencies correspond to 2
  variables one for rows, one for columns.
• ? often used to analyze survey results
• ? used to determine whether 2 variables are
  independent of each other

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• A test of independence tests the null hypothesis
  that the row variable and the column variable in
  a contingency table are not related.

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Assumptions

1. The sample data are randomly selected.
2. The null hypothesis is the statement that the row
   and column variables are independent the
   alternative hypothesis is the statement that the
   row and column variables are dependent.
3. For every cell in the frequency table, the
   expected frequency E is at least 5.

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• The test will follow the traditional method,
  with the following test statistic
• The critical values are found in Table A-4 with
  degrees of freedom ( rows 1)( cols 1)
  the critical region is located in the right tail
  only.

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Example 1

• A study was conducted of 531 persons injured in
  bicycle crashes, and randomly selected sample
  results are summarized in the following table.
  At the 0.05 significance level, test the claim
  that wearing a helmet has no effect on whether
  facial injuries are received. Based on these
  results, does a helmet seem to be effective in
  helping to prevent facial injuries in a crash?

Helmet Worn No Helmet
Facial Injuries 30 182
Nonfacial Injuries 83 236
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First, find the expected frequencies
Helmet Worn No Helmet Total
Facial Injuries 30 45.11 182 166.89 212
Non-facial Injuries 83 67.89 236 251.11 319
Total 113 418 531
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• HO Wearing a helmet and receiving facial
  injuries are independent.
• H1 Receiving facial injuries depends on
  whether or not you are wearing a helmet.

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Test Statistic
O E
30 83 182 236 45.11 67.89 166.89 251.11 5.06 1.37 3.36 0.91
10.7
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• Reject Ho!
• The data indicates that wearing a helmet and
  receiving facial injuries are related.

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• For a test of homogeneity (having the same
  quality), we test the claim that different
  populations have the same proportions of some
  characteristics.
• The only difference between it and the test of
  independence is the statement of the null and
  alternative hypothesis
• ? these are listed as The proportions are the
  same (be specific to the problem), The
  proportions are different.

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Example 2

• A health club manager wants to determine whether
  the number of days per week that college students
  spend exercising is related to gender. A random
  sample of 275 students is selected and the
  results are shown in the following table

Days per week spent exercising Days per week spent exercising Days per week spent exercising Days per week spent exercising
Gender 0-1 2-3 4-5 6-7
Male 40 53 26 6
Female 34 68 37 11
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• At a0.05, is there enough evidence to conclude
  that the number of days spent exercising per week
  is related to gender??

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H O M O G E N E I T Y