1. Independent Samples: Comparing Proportions
Lecture 49
Section 11.5
Wed, Apr 21, 2004

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 difference in proportions could be attributed to that one difference.

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
1 = (p1(1 – p1)/n1).
Similarly, p2^ is N(p2, 2), where
2 = (p2(1 – p2)/n2).
Therefore, p1^ – p2^ is
N(p1 – p2, (12 + 22).

7. The Standard Deviation of p1^ – p2^

8. 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).

9. 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.

10. Pooled Estimate of p

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

12. Caution If the null hypothesis does not say H0: p1 = p2
then we should not use the pooled estimate p^.

13. 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.

14. 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?

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

16. Hypothesis Testing
Compute the test statistic.

17. Hypothesis Testing
First, we must compute p^.

18. Hypothesis Testing
Now we can compute z.

19. Hypothesis Testing Compute the p-value. State the conclusion.
p-value = normalcdf(-99, )
= 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.”

20. 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.

21. Assignment