Section 13.2 Chi-Squared Test of Independence/Association.
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Section 13.2 Section 13.2 Chi-Squared Test of Chi-Squared Test of Independence/Association Independence/Association
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What are the assumptions? SRS from the population of interest 10% Rule for showing independent. None of the expected counts are less than 5.
Transcript of Section 13.2 Chi-Squared Test of Independence/Association.
Section 13.2Section 13.2
Chi-Squared Test of Chi-Squared Test of Independence/AssociationIndependence/Association
The Three Types of χ2 tests
• Recall that there are three types of χ2
tests:– Goodness of fit test
• Tests whether a sample distribution matches a hypothesized distribution (Does it fit GOOD?)
– Homogeneity of proportions• Tests whether p1 = p2 = p3 = …
– Association/Independence• Tests whether two categorical variables are
related
What are the assumptions?
• SRS from the population of interest
• 10% Rule for showing independent.
• None of the expected counts are less than 5.
χ2 test of Association• When a two-way table compares
several populations, we use a χ2 test for homogeneity of proportions.
• On the other hand, sometimes we want to classify observations FROM A SINGLE POPULATION by placing them in categories. Then we use a χ2 test of association (also called the χ2 test of independence).
ExampleSES
Smoking High Middle Low TotalCurrent 51 22 43 116Former 92 21 28 141Never 68 9 22 99Total 211 52 93 356
My question: “Is there an association between SES level and smoking habits?”
Hypotheses• The χ2 test of association has
hypotheses that are written in words:– H0: There is no relationship between
smoking and SES level.– Ha: There is some relationship
between smoking and SES level.
In other words…– H0: Smoking and SES level are independent.– Ha: Smoking and SES level are not
independent.OR
– H0: There is no association between smoking and SES level.
– Ha: There is some association between smoking and SES level.
Finishing up• The rest of the test is carried out
in the same manner as the χ2 test for homogeneity of proportions.
• Just make sure your conclusion sentences match the H0 and Ha hypotheses.
χ2 tests Comparison Chart
Goodness of Fit Homogeneity of
Proportions
Association
Hypotheses H0: The sample distribution matches the hypothesized distribution.Ha: The sample distribution does not match the hypothesized distribution.
H0: p1 = p2 = p3 = …Ha: Not all of the proportions are equal.
H0: There is no relationship between the two variables.Ha: There is some relationship between the two variables.
Assumptions
SRSNone of the expected counts are zero.No more than 20% of the expected counts are less than 5.
Independent SRSsNone of the expected counts are zero.No more than 20% of the expected counts are less than 5.
SRSNone of the expected counts are zero.No more than 20% of the expected counts are less than 5.
Test Multiply the expected percents by your sample size. Calculate χ2 statistic by hand. Look up p-value in Table E, df = # of categories minus 1.
Enter observed counts in [A]. Use the calculator to find the p-value. *LOOK AT THE EXPECTED COUNTS IN [B] TO CHECK ASSUMPTIONS!
Enter observed counts in [A]. Use the calculator to find the p-value. *LOOK AT THE EXPECTED COUNTS IN [B] TO CHECK ASSUMPTIONS!
So, the difference between the homogeneity of proportions test and the association test is ARE WE COMPARING SEVERAL POPULATIONS or DID THE DATA ARISE BY CLASSIFYING OBSERVATIONS INTO CATEGORIES.
HomeworkHomeworkChapter 11Chapter 11
# 48, 49, 50, 51, 52# 48, 49, 50, 51, 52
