1. Statistics 8.4 8.4 Testing the Difference Between Proportions Statistics Mrs. Spitz Spring 2009

2. Statistics 8.4 Objectives/Assignment How to perform a z-test for the difference between two population proportions p 1 and p 2. Assignment: pp. 404-406 #1-12

3. Statistics 8.4

4. Two Sample z-Test for the Difference Between Proportions In this section, you will learn how to use a z- test to test the difference between two population proportions p 1 and p 2 using a sample proportion from each population. For instance, suppose you want to determine whether the proportion of female college students who earn a bachelor’s degree in four years is different from the proportion of male college students who earn a bachelor’s degree in four years.

5. Statistics 8.4 Two Sample z-Test for the Difference Between Proportions To use a z-test to test such a difference, the following conditions are necessary: 1. The samples must be independent. 2. The samples must be large enough to use a normal sampling distribution. That is: n 1 p 1 ≥ 5, n 1 q 1 ≥ 5, n 2 p 2 ≥ 5, n 2 q 2 ≥ 5

6. Statistics 8.4 Two Sample z-Test for the Difference Between Proportions If these conditions are met, then the sampling distribution for the difference between the sample proportions, is a normal distribution with mean: And a standard error

7. Statistics 8.4 Two sample z-test for difference between proportions Notice that you need to know the population proportions to calculate the standard error. Because a hypothesis test for p 1 - p 2 is based on the condition of equality that p 1 = p 2, you can calculate a weighted estimate of p using: and where

8. Statistics 8.4 Two-Sample z-Test for the Difference Between Proportions Using the weighted estimate, the standard error of the sampling distribution for is: where Also, when determining whether the z-test can be used for the difference between proportions, you should use in place of p 1 and p 2 and in place of q 1 and q 2

9. Statistics 8.4 Two Sample z-Test for the Difference Between Proportions If the sampling distribution for is normal, you can use a two-sample z-test to test the difference between two population proportions, and

10. Statistics 8.4 If the null hypothesis states that p 1 = p 2, then the expression, p 1 - p 2 is equal to 0 in the preceding test.

11. Statistics 8.4 Ex. 1: A Two-Sample z-Test for the Difference Between Proportions In a study of 200 adult female and 250 adult male Internet users, 30% of the females and 38% of the males said that they plan to shop online at least once during the next month. At  = 0.10, test the claim that there is a difference in the proportion of female Internet users who plan to shop online and the proportion of male Internet users who plan to shop online.

12. Statistics 8.4 Solution You want to determine whether there is a difference in the proportions. So, the null and alternative hypotheses are: H o : p 1 = p 2 and p 1  p 2 (claim) Because the test is two-tailed and the level of significance is  = 0.10, the critical values are  1.645. The rejection regions are z 1.645.

13. Statistics 8.4 Solution The weighted estimate of the population proportion is: And. Because 200(0.344), 200(0.656), 250(0.344) and 250)0.656 are at least 5, you can use the two-sample z-test.

14. Statistics 8.4 Solution The standardized test statistic is:

15. Statistics 8.4 Solution The graph shows the location of the rejection regions and the standardized test statistic. Because z is in the rejection region, you should decide to reject the null hypothesis. You have enough evidence at the 10% level to conclude there is a difference in the proportion of female and the proportion of male Internet users who plan to shop online.

16. Statistics 8.4 Ex. 2: Two-Sample z-Test for the Difference Between Proportions A medical research team conducted a study to test the effect of a cholesterol-reducing medication. At the end of the study, the researchers found that of the 4700 subjects who took the medication, 301 died of heart disease. Of the 4300 subjects who took a placebo, 357 died of heart disease. At  = 0.01, can you conclude that the death rate is lower for those who took the medication than for those who took the placebo?

17. Statistics 8.4 Solution You want to determine whether there is a difference in the proportions. So, the null and alternative hypotheses are: H o : p 1 ≥ p 2 and p 1 < p 2 (claim) Because the test is left-tailed and the level of significance is  = 0.01, the critical value is -2.33. The rejection region is z < -2.33

18. Statistics 8.4 Solution The weighted estimate of the population proportion is: And. Because 4700(0.073), 4700(0.927), 4300(0.073) and 4300)0.927 are at least 5, you can use the two-sample z-test.

19. Statistics 8.4 Solution The standardized test statistic is:

20. Statistics 8.4 Solution The graph shows the location of the rejection region and the standardized test statistic. Because z is in the rejection region, you should decide to reject the null hypothesis. At the 1% level, there is enough evidence to conclude that the death rate is lower for those who took the medication than for those who took the placebo.