Independent Samples: Comparing Proportions

1. Independent Samples: Comparing Proportions Lecture 38 Section 11.5 Wed, Nov 7, 2007

2. The Sampling Distribution of p1^ – p2^ • p1^ is N(p1, 1), where • p2^ is N(p2, 2), where

3. Statistical Fact #1 • For any two random variables X and Y,

4. Statistical Fact #2 • Furthermore, if X and Y are both normal, then X – Y is normal. • That is, if X is N(X, X) and Y is N(Y, Y), then

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

6. The Test Statistic • Therefore, the test statistic would be if we knew the values of p1 and p2. • We will approximate them with p1^ and p2^.

7. The Test Statistic • Therefore, the test statistic would be except that…

8. Pooled Estimate of p • The null hypothesis is typically H0: p1 = p2 • Under that assumption, p1^ and p2^ are both estimators of a common value, which we will call p.

9. Pooled Estimate of p • Rather than use either p1^ or p2^ 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.

10. Pooled Estimate of p • The “Batting-Average” Formula:

11. Case Study 13 • In the survey, we had 240 males (48%) and 260 females (52%). • 41% of the males, or 98 males, said Wilder is doing good or excellent. • 37% of the females, or 96 females, said he is doing good or excellent. • Altogether, 194 people out of 500, or 38.8%, said he is doing good or excellent.

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

13. Case Study 13 • Compute

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

15. The Test Statistic • So the test statistic is where

16. The Value of the Test Statistic • Compute z:

17. The p-value, etc. • Compute the p-value: P(Z > 0.9170) =0.1796. • Reject H0. • Equal proportions of men and women believe that Mayor Wilder is doing a good or excellent job.

18. Case Study 13 Continued • Do equal proportions of whites and blacks believe that Mayor Wilder is doing a good or excellent job? • Do equal proportions of Republicans and Democrats believe that Mayor Wilder is doing a good or excellent job? • City Hall turmoil: RT-D poll.

19. TI-83 – Testing Hypotheses Concerning p1^ – p2^ • Press STAT > TESTS > 2-PropZTest... • Enter • x1, n1 • x2, n2 • Choose the correct alternative hypothesis. • Select Calculate and press ENTER.

20. TI-83 – Testing Hypotheses Concerning p1^ – p2^ • In the window the following appear. • The title. • The alternative hypothesis. • The value of the test statistic z. • The p-value. • p1^. • p2^. • The pooled estimate p^. • n1. • n2.

21. Example • Work Case Study 13 again, using the TI-83.

22. Confidence Intervals for p1^ – p2^ • The formula for a confidence interval for p1^ – p2^ is • Caution: Note that we do not use the pooled estimate for p^.

23. TI-83 – Confidence Intervals for p1^ – p2^ • Press STAT > TESTS > 2-PropZInt… • Enter • x1, n1 • x2, n2 • The confidence level. • Select Calculate and press ENTER.

24. TI-83 – Confidence Intervals for p1^ – p2^ • In the window the following appear. • The title. • The confidence interval. • p1^. • p2^. • n1. • n2.

25. Example • Find a 95% confidence interval for the difference between the proportions of whites and blacks who believe that Mayor Wilder is doing a good or excellent job.