1. Independent Samples: Comparing Proportions Lecture 35 Section 11.5 Mon, Nov 20, 2006

2. Comparing Proportions We now wish to compare proportions between two populations. We now wish to compare proportions between two populations. Normally, we would be measuring proportions for the same attribute. Normally, we would be measuring proportions for the same attribute. For example, we could measure the proportion of NC residents living below the poverty level and the proportion of VA residents living below the poverty level. For example, we could measure the proportion of NC residents living below the poverty level and the proportion of VA residents living below the poverty level.

3. Examples The “gender gap” – the proportion of men who vote Republican vs. the proportion of women who vote Republican. The “gender gap” – the proportion of men who vote Republican vs. the proportion of women who vote Republican. The proportion of teenagers who smoked marijuana in 1995 vs. the proportion of teenagers who smoked marijuana in 2000. The proportion of teenagers who smoked marijuana in 1995 vs. the proportion of teenagers who smoked marijuana in 2000.

4. Examples The proportion of patients who recovered, given treatment A vs. the proportion of patients who recovered, given treatment B. The proportion of patients who recovered, given treatment A vs. the proportion of patients who recovered, given treatment B. Treatment A could be a placebo. Treatment A could be a placebo.

5. Comparing proportions To estimate the difference between population proportions p 1 and p 2, we need the sample proportions p 1 ^ and p 2 ^. To estimate the difference between population proportions p 1 and p 2, we need the sample proportions p 1 ^ and p 2 ^. The difference p 1 ^ – p 2 ^ is an estimator of the difference p 1 – p 2. The difference p 1 ^ – p 2 ^ is an estimator of the difference p 1 – p 2.

6. Hypothesis Testing See Example 11.8, p. 721 – Perceptions of the U.S.: Canadian versus French. See Example 11.8, p. 721 – Perceptions of the U.S.: Canadian versus French. p 1 = proportion of Canadians who feel positive about the U.S.. p 1 = proportion of Canadians who feel positive about the U.S.. p 2 = proportion of French who feel positive about the U.S.. p 2 = proportion of French who feel positive about the U.S..

7. Hypothesis Testing The hypotheses. The hypotheses. H0: p 1 – p 2 = 0 (i.e., p 1 = p 2 ) H0: p 1 – p 2 = 0 (i.e., p 1 = p 2 ) H1: p 1 – p 2 > 0 (i.e., p 1 > p 2 ) H1: p 1 – p 2 > 0 (i.e., p 1 > p 2 ) The significance level is  = 0.05. The significance level is  = 0.05. What is the test statistic? What is the test statistic? That depends on the sampling distribution of p 1 ^ – p 2 ^. That depends on the sampling distribution of p 1 ^ – p 2 ^.

8. The Sampling Distribution of p 1 ^ – p 2 ^ If the sample sizes are large enough, then p 1 ^ is N(p 1,  1 ), where If the sample sizes are large enough, then p 1 ^ is N(p 1,  1 ), where Similarly, p 2 ^ is N(p 2,  2 ), where Similarly, p 2 ^ is N(p 2,  2 ), where

9. The Sampling Distribution of p 1 ^ – p 2 ^ The sample sizes will be large enough if The sample sizes will be large enough if n 1 p 1  5, and n 1 (1 – p 1 )  5, and n 1 p 1  5, and n 1 (1 – p 1 )  5, and n 2 p 2  5, and n 2 (1 – p 2 )  5. n 2 p 2  5, and n 2 (1 – p 2 )  5.

10. The Sampling Distribution of p 1 ^ – p 2 ^ Therefore, Therefore,where

11. The Test Statistic Therefore, the test statistic would be Therefore, the test statistic would be if we knew the values of p 1 and p 2. We could estimate them with p 1 ^ and p 2 ^. We could estimate them with p 1 ^ and p 2 ^. But there may be a better way… But there may be a better way…

12. Pooled Estimate of p In hypothesis testing for the difference between proportions, typically the null hypothesis is In hypothesis testing for the difference between proportions, typically the null hypothesis is H 0 : p 1 = p 2 Under that assumption, p 1 ^ and p 2 ^ are both estimators of a common value (call it p). Under that assumption, p 1 ^ and p 2 ^ are both estimators of a common value (call it p).

13. Pooled Estimate of p Rather than use either p 1 ^ or p 2 ^ alone to estimate p, we will use a “pooled” estimate. Rather than use either p 1 ^ or p 2 ^ alone to estimate p, we will use a “pooled” estimate. The pooled estimate is the proportion that we would get if we pooled the two samples together. The pooled estimate is the proportion that we would get if we pooled the two samples together.

14. Pooled Estimate of p (The “Batting-Average” Formula)

15. The Standard Deviation of p 1 ^ – p 2 ^ This leads to a better estimator of the standard deviation of p 1 ^ – p 2 ^. This leads to a better estimator of the standard deviation of p 1 ^ – p 2 ^.

16. Caution If the null hypothesis does not say If the null hypothesis does not say H 0 : p 1 = p 2 then we should not use the pooled estimate p ^, but should use the unpooled estimate

17. The Test Statistic So the test statistic is So the test statistic is

18. The Value of the Test Statistic Compute p ^ : Compute p ^ : Now compute z: Now compute z:

19. The p-value, etc. Compute the p-value: Compute the p-value: P(Z > 7.253) = 2.059  10 -13. Reject H 0. Reject H 0. The data indicate that a greater proportion of Canadians than French have a positive feeling about the U.S. The data indicate that a greater proportion of Canadians than French have a positive feeling about the U.S.

20. Exercise 11.34 Sample #1: 361 “Wallace” cars reveal that 270 have the sticker. Sample #1: 361 “Wallace” cars reveal that 270 have the sticker. Sample #2: 178 “Humphrey” cars reveal that 154 have the sticker. Sample #2: 178 “Humphrey” cars reveal that 154 have the sticker. Do these data indicate that p 1  p 2 ? Do these data indicate that p 1  p 2 ?

21. Example State the hypotheses. State the hypotheses. H 0 : p 1 = p 2 H 0 : p 1 = p 2 H 1 : p 1  p 2 H 1 : p 1  p 2 State the level of significance. State the level of significance.  = 0.05.  = 0.05.

22. Example Write the test statistic. Write the test statistic.

23. Example Compute p 1 ^, p 2 ^, and p ^. Compute p 1 ^, p 2 ^, and p ^.

24. Example Now we can compute z. Now we can compute z.

25. Example Compute the p-value. Compute the p-value. p-value = 2  normalcdf(-E99, -3.126) p-value = 2  normalcdf(-E99, -3.126) = 2(0.0008861) = 0.001772. The decision is to reject H 0. The decision is to reject H 0. State the conclusion. State the conclusion. The data indicate that the proportion of Wallace cars that have the sticker is different from the proportion of Humphrey cars that have the sticker. The data indicate that the proportion of Wallace cars that have the sticker is different from the proportion of Humphrey cars that have the sticker.

26. TI-83 – Testing Hypotheses Concerning p 1 ^ – p 2 ^ Press STAT > TESTS > 2-PropZTest... Press STAT > TESTS > 2-PropZTest... Enter Enter x 1, n 1 x 1, n 1 x 2, n 2 x 2, n 2 Choose the correct alternative hypothesis. Choose the correct alternative hypothesis. Select Calculate and press ENTER. Select Calculate and press ENTER.

27. TI-83 – Testing Hypotheses Concerning p 1 ^ – p 2 ^ In the window the following appear. In the window the following appear. The title. The title. The alternative hypothesis. The alternative hypothesis. The value of the test statistic z. The value of the test statistic z. The p-value. The p-value. p 1 ^. p 1 ^. p 2 ^. p 2 ^. The pooled estimate p^. The pooled estimate p^. n 1. n 1. n 2. n 2.

28. Example Work Exercise 11.34 using the TI-83. Work Exercise 11.34 using the TI-83.

29. Confidence Intervals for p 1 ^ – p 2 ^ The formula for a confidence interval for p 1 ^ – p 2 ^ is The formula for a confidence interval for p 1 ^ – p 2 ^ is Caution: Note that we do not use the pooled estimate for p ^. Caution: Note that we do not use the pooled estimate for p ^.

30. TI-83 – Confidence Intervals for p 1 ^ – p 2 ^ Press STAT > TESTS > 2-PropZInt… Press STAT > TESTS > 2-PropZInt… Enter Enter x 1, n 1 x 1, n 1 x 2, n 2 x 2, n 2 The confidence level. The confidence level. Select Calculate and press ENTER. Select Calculate and press ENTER.

31. TI-83 – Confidence Intervals for p 1 ^ – p 2 ^ In the window the following appear. In the window the following appear. The title. The title. The confidence interval. The confidence interval. p 1 ^. p 1 ^. p 2 ^. p 2 ^. n 1. n 1. n 2. n 2.

32. Example Find a 95% confidence interval using the data in Exercise 11.34. Find a 95% confidence interval using the data in Exercise 11.34.