1. Comparing Two Proportions
Chapter 22
Comparing Two Proportions

2. Topics
Finding a Confidence Interval to represent the difference between two proportions.
Testing to determine if there is a statistically significant difference between two proportions

3. Comparing Two Proportions
Suppose an experiment is performed comparing SAT test scores using an SAT prep class and test scores without a training class. In a randomly selected high school, 250 juniors are assigned to a prep course and 240 juniors are assigned to no prep course. 73% of the students who took the prep class scored above % of the students who didn’t take the course scored above Is there evidence to suggest that the SAT prep course improved those students scores?

4. Test Preparation Example
What is the sample proportion for each group?
Do these proportions differ in the direction conjectured by the researchers?

5. Test Example ctd.
Even if there were absolutely no effect of the observer’s interest, is it possible to have gotten such a big difference between the groups due to sample variation?

6. Significance Test of Equality of p1 and p2
Test Statistic:
Where n1 and n2 are the respective sample sizes, p1 and p2 are the respective sample proportions, and pc is the pooled sample proportion (the proportion of “successes” if the two samples were pooled together).

7. Conditions for validity of the test
The two samples are independently selected simple random samples from the population of interest

8. Summary
We say a sample result is statistically significant if it is unlikely to occur by chance (i.e. due to normal variability in the sample).
In this problem, we have approximated the P-value (the likelihood that a result at least this extreme would occur by chance due to sample variability).

9. Confidence Interval for p1 - p2
Once we have determined if there is a statistically significant difference between the two population proportions, we may be interested in studying how big this difference is.
Again, we only have sample data to compare, but we can create a confidence interval of values that the actual difference is likely to lie between.

10. Confidence Interval for p1 - p2
A confidence interval for p1 - p2 is given by:
The validity conditions are the same as for the test of significance.

11. Example
In 1991, researchers at the National Cancer Institute released the results of a study that investigated the effect of weed-killing herbicides on house pets. They examined 827 dogs from homes where an herbicide was used on a regular basis, diagnosing malignant lymphoma in 473 of them. Of the 130 dogs from homes where no herbicides were used, only 19 were found to have lymphoma.

12. Example ctd. What are the two sample proportions of interest?
What’s the standard error of the difference in the two proportions?

13. Example ctd.
Perform a test of significance to determine whether there is a “statistically significant” difference between the cancer rates.

14. Example ctd.
Find a 95% confidence interval for the difference between the rates.

15. Example ctd.
Interpret the result:

16. Example
A poll checking on the level of public support for proposed antiterrorist legislation reported that 68% of the respondents were in favor. The pollsters reported a sampling error of ±3%. When the responses were broken down by party affiliation, support was 2% higher among Republican respondents than Democrats. The pollsters said the margin of error for this difference was ±4%.

17. Example ctd.
Why is the margin of error larger for the difference in support between the parties than for the overall level of support?
Based on these results, can we conclude that support is significantly higher among Republicans?