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 not use the pooled estimate p^, but should use the unpooled 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.