1.  2 test for independence Used with categorical, bivariate data from ONE sample Used to see if the two categorical variables are associated (dependent) or not associated (independent)

2. Assumptions & formula remain the same!

3. Hypotheses – written in words H 0 : two variables are independent H a : two variables are dependent Be sure to write in context!

4. Example 5: A beef distributor wishes to determine whether there is a relationship between geographic region and cut of meat preferred. If there is no relationship, we will say that beef preference is independent of geographic region. Suppose that, in a random sample of 500 customers, 300 are from the North and 200 from the South. Also, 150 prefer cut A, 275 prefer cut B, and 75 prefer cut C.

5. If beef preference is independent of geographic region, how would we expect this table to be filled in? NorthSouthTotal Cut A150 Cut B275 Cut C75 Total300200500 9060 165110 4530

6. Expected Counts Assuming H 0 is true,

7. Degrees of freedom Or cover up one row & one column & count the number of cells remaining!

8. Now suppose that in the actual sample of 500 consumers the observed numbers were as follows: Is there sufficient evidence to suggest that geographic regions and beef preference are not independent? (Is there a difference between the expected and observed counts?) NorthSouth Cut A10050 Cut B168107 Cut C3243

9. Assumptions: Have a random sample of people All expected counts are greater than 5. H 0 : geographic region and beef preference are independent H a : geographic region and beef preference are dependent P-value =.0021df = 2  =.05 Since p-value < , I reject H 0. There is sufficient evidence to suggest that geographic region and beef preference are dependent. Expected Counts: N S A90 60 B165110 C45 30

10. Example 6 : Suppose that residents of a particular city can watch national news on affiliates of 4 different networks: ABC, CBS, PBS and NBC. A researcher wants to know whether there is any relationship between political philosophies (liberal, moderate or conservative) and the network they watch for news. A random sample of 300 viewers was selected, and the results are given below: SURVEY RESULTS ABCCBSNBCPBS Liberal20 2515 Moderate45355020 Conservative1540105

11. Each observation consists of. We would like to test whether there is. 1. State null and alternate hypothesis

12. ASSUMPTIONS:

13. EXPECTED COUNTS ABCCBSNBCPBS Liberal Moderate Conservative 2. Calculate statistics: Expected counts: Degrees of freedom: Use the graphing calculator to calculate P-value.

14. 3. State conclusion.

15.  2 test for homogeneity single categorical two (or more) independent samplesUsed with a single categorical variable from two (or more) independent samples Used to see if the two populations are the same (homogeneous)

16. Assumptions & formula remain the same! Expected counts & df are found the same way as test for independence. Only Only change is the hypotheses!

17. Hypotheses – written in words H 0 : the two (or more) distributions are the same H a : the distributions are different Be sure to write in context!

18. Separate random samples of 124 consumers and 101 dentists were asked to respond to the following statement: “I favor the use of advertising by dentists to attract new patients.” Possible responses were strongly agree, agree, neutral, disagree, and strongly disagree. The authors were interested in determining whether the two groups – consumers and dentists – differed in their attitudes towards advertising. SURVEY RESPONSES Strongly Agree AgreeNeutralDisagree Strongly Disagree Dentists3449945 Consumers918232846

19. Assumptions: Have 2 random samples. All expected counts are greater than 5. H 0 : the true proportions are the same for both groups. H a : the true proportions differ in at least one category P-value =.000df = 4  =.05 Since p-value < , I reject H 0. There is sufficient evidence to conclude that the true proportion for the 2 groups differ. Expected Counts Strongly Agree AgreeNeutralDisagree Strongly Disagree Dentists19.330.114.4 22.9 Consumers23.736.917.6 28.1

20. From pollingreport.com: CNN/Opinion Research Corporation Poll. June 16, 2010. N=500 adults nationwide. MoE ± 4. "How do you feel about increased drilling for oil and natural gas offshore in U.S. waters? Do you strongly favor, mildly favor, mildly oppose or strongly oppose increased offshore drilling?" Strongly Favor Mildly Favor Mildly Oppose Strongly Oppose June, 201013011585170 July, 200822511560100