1. 10-1

2. 10-2 Chapter Ten Comparing Proportions and Chi-Square Tests McGraw-Hill/Irwin Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved.

3. 10-3 Comparing Proportions and Chi-Square tests 10.1 Comparing Two Population Proportions 10.2 The Chi-Square Distribution 10.3 Chi-Square Goodness of Fit Tests 10.4A Chi-Square Test for Independence

4. 10-4 Large Sample Interval for the Difference in Proportions If two independent samples are both large, a 100(1 -  )% confidence interval for p 1 - p 2 is 10.1 Comparing Two Population Proportions

5. 10-5 Large Sample Test for Difference in Proportions Test Statistics If two sampled populations are both large, we can reject H 0 : p 1 - p 2 = D 0 at the  level of significance if and only if the appropriate rejection point condition holds or, equivalently, if the corresponding p-value is less than . Alternative Reject H 0 if:p-Value

6. 10-6 Example: Difference Between Proportions: Interval and Test Test H 0 : p 1 - p 2 = 0 versus H a : p 1 - p 2  0 Example 10.2 Advertising Media 95% Confidence Interval for p 1 - p 2

7. 10-7 10.2 The Chi-Square Distribution The chi-square distribution depends on the number of degrees of freedom. A chi-square point is the point under a chi-square distribution that gives right-hand tail area .

8. 10-8 10.3 Chi-Square Goodness of Fit Test Example 10.4 The Microwave Oven Preference Case Are consumer preferences for microwave ovens in Milwaukee the same as those historically observed in Cleveland? MegaStat Output

9. 10-9 A Goodness of Fit Test for Multinomial Probabilities Consider the outcome of a multinomial experiment where each of n randomly selected items is classified into one of k groups and let f i = number of items classified into group i (ith observed frequency) E i = np i = expected number in ith group if p i is probability of being in group i (ith expected frequency) H 0 : multinomial probabilities are p 1, p 2, …, p k H a : at least one of the probabilities differs from p 1, p 2, …, p k Test Statistic: Reject H 0 if   >     or if p-value <  To Test:  2 and the p-value are based on p-1 degrees of freedom. Values of  2 are given in Table A.17.

10. 10-10 Example: Chi-Square Goodness of Fit Test Example 10.4 The Microwave Oven Preference Case H 0 : p 1 =.20, p 2 =.35, p 3 =.30, p 4 =.15 H a : H 0 fails to hold

11. 10-11 Chi-Square Goodness of Fit for Normal Distribution Example 10.5 The Car Mileage Case H 0 : car mileage data are random sample from normal population H a : data not from a normal population     and the p-value are based on k-1-m = 6-1-2 = 3 degrees of freedom.

12. 10-12 10.4 Chi-Square Test for Independence Example 10.6 The Client Satisfaction Case Does investment client satisfaction depend upon investment fund type? MegaStat Output

13. 10-13 A Chi-Square Test for Independence Test Statistic: Reject H 0 if   >     or if p-value <      and the p-value are based on (r-1)(c-1) degrees of freedom. Values of    are given in Table A.17. H 0 : the two classifications statistically independent H a : the two classifications statistically dependent To Test: Each of n randomly selected items is classified on two dimensions into a contingency table with r rows an c columns and let f ij = observed cell frequency for ith row and jth column r i = ith row total, c j = jth column total expected cell frequency for ith row and jth column under independence

14. 10-14 Example: Chi-Square Test for Independence Example 10.6 The Client Satisfaction Case H 0 : client satisfaction is independent of fund type H a : client satisfaction depends upon fund type Client Satisfaction Fund High Low Med All Bond 15 3 12 30 12 6 12 Stock 24 2 4 30 12 6 12 TaxDef 1 15 24 40 16 8 16 All 40 20 40 100

15. 10-15 Example: Analysis of Classification Dependencies Example 10.7 The Client Satisfaction Case Client Satisfaction Fund High Low Med All Bond 15 3 12 30 50.00 10.00 40.00 100.00 Stock 24 2 4 30 80.00 6.67 13.33 100.00 TaxDef 1 15 24 40 2.50 37.50 60.00 100.00 Row Percentages Row Percentages versus Investment Type for each Satisfaction Level

16. 10-16 Comparing Proportions and Chi-Square tests Summary: 10.1 Comparing Two Population Proportions 10.2 The Chi-Square Distribution 10.3Chi-Square Goodness of Fit Tests 10.4A Chi-Square Test for Independence