Independent Samples: Comparing Proportions

1. Independent Samples: Comparing Proportions Lecture 41 Section 11.5 Tue, Apr 5, 2005

2. Comparing Proportions • We now wish to compare proportions between two populations. • The populations should be similar in most respects. • Ideally, they would differ in only one respect. • Any significant difference in proportions could be attributed to that one difference in characteristic.

3. Examples • 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.

4. Examples • 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.

5. Comparing proportions • To estimate the difference between population proportions p1 and p2, we need the sample proportions p1^ and p2^. • The difference p1^ – p2^ is an estimator of the difference p1 – p2. • What is the sampling distribution of p1^ – p2^?

6. The Sampling Distribution of p1^ – p2^ • If the sample sizes are large enough, then p1^ is N(p1, 1), where • Similarly, p2^ is N(p2, 2), where

7. The Sampling Distribution of p1^ – p2^ • Therefore, where

8. The Sampling Distribution of p1^ – p2^ • The sample sizes will be large enough if • n1p1 5, and n1(1 – p1)  5, and • n2p2 5, and n2(1 – p2)  5.

9. Pooled Estimate of p • In hypothesis testing for the difference between proportions, typically the null hypothesis is H0: p1 = p2 • Under that assumption, p1^ and p2^ are both estimators of a common value (call it p).

10. Pooled Estimate of p • Rather than use either p1^ or p2^ alone to estimate p, we will use a “pooled” estimate. • That is the proportion that we would get if we pooled the two samples together.

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

12. The Standard Deviation of p1^ – p2^ • This leads to a better estimator of the standard deviation of p1^ – p2^.

13. Caution • If the null hypothesis does not say H0: p1 = p2 then we should not use the pooled estimate p^, but should use the unpooled estimate

14. Hypothesis Testing • See Example 11.8, p. 669 – Feeling Successful: Women versus Men. • p1 = proportion of women who say that a paycheck makes them feel successful. • p2 = proportion of men who say that a paycheck makes them feel successful.

15. Hypothesis Testing • A sample of 1001 women shows that 7% agree. • A sample of 460 men shows that 26% agree. • Do these data demonstrate that the proportion is lower for women?

16. Hypothesis Testing • State the hypotheses. • H0: p1 = p2 • H1: p1 < p2 • State the level of significance. •  = 0.05.

17. Hypothesis Testing • Compute the test statistic.

18. Hypothesis Testing • First, we must compute p^.

19. Hypothesis Testing • Now we can compute z.

20. Hypothesis Testing • Compute the p-value. • p-value = normalcdf(-E99, -10.03) = 0. • State the conclusion. • “We conclude that the proportion of women who say that a paycheck makes them feel successful is less than the proportion of men who say that.”

21. Let’s Do It! • Let’s do it! 11.7, p. 671 – HMOs on the Rise. • Test the hypothesis that p1 = p2 vs. p1 > p2, where • p1 = proportion in the South. • p2 = proportion in the North.