CHAPTER 11Chi-Square Tests

11.3 Contingency Tables Two way classification table – presents

information on more than one variable for each element

Example: college students broken into gender and enrollment status

11.4.1 Test of Independence

Null Hypothesis: the two attributes (characteristics) of the elements are not related (independent)

Alternative Hypothesis: the two attributes of the elements are related (dependent)

Example: We could test to see if there was an association between being a man or woman and having a preference for watching sports or soap operas

Formulas Degrees of Freedom: df = (R-1)(C-1)

where R is the number of rows and C is the number of columns in your table

Test Statistic: where the observed and expected frequencies are calculated for each cell

Expected frequency:

EEO 2

2 )(

SampleSizelColumnTotaRowTotalE ))((

Example #1 11-6 Pg 503 Violence and lack of discipline have become

major problems in schools in the US. A random sample of 300 adults was selected and these adults were asked if they favor giving more freedom to schoolteachers to punish students for violence and lack of discipline. The two-way classification of the responses of these adults is presented in the following table. Does the sample provide sufficient evidence to conclude that the two attributes, gender and opinions of adults, are dependent? Use a 1% significance level.

Prem Mann, Introductory Statistics, 6/ECopyright 2007 John Wiley & Sons. All rights reserved.

Table 11.6 (p. 504)

Answers1. H₀: Gender and opinions of adults are independent.

H₁: Gender and opinions of adults are dependent.

2. Chi-Square distribution because we are testing for independence with a contingency table.

3. Rejection/Non-rejection Regions: χ² = 9.2104. Test Statistic: χ² = 8.2525. Since our test statistic falls in our non-rejection region,

the two characteristics of gender and opinion on allowing school teachers more freedom for punishment are dependent.

11.4.2 Test of Homogeneity

Null Hypothesis: two more populations are homogeneous (similar) with regard to the distribution of a certain characteristic

Example: Are the preferences of people in Florida, Arizona, and Vermont similar with regards to Pepsi, Coke, or 7-UP?

Example #2 11-8 Pg 509 After taking a sample of 250 households

from California and 150 households from Wisconsin, the following table was constructed showing three different income groups. Using the 2.5% significance level, test the null hypothesis that the distribution of households with regard to income levels is similar (homogeneous) for the two states.

Prem Mann, Introductory Statistics, 6/ECopyright 2007 John Wiley & Sons. All rights reserved.

Table 11.8 (p. 506)

Answers1. H₀: The proportions of households that belong to

different income groups are the same in both states. H₁: The proportions of households that belong to different income groups are not the same in both states.

2. Chi-square because we are testing for homogeneity3. Rejection/Non-Rejection regions: χ² = 7.3784. Test Statistic: χ² = 4.3395. Since the test statistic falls in the non-rejection region,

the proportions are similar in California and Wisconsin.

Homework Page #511 #23-26, 30