1. Chapter 22: Comparing Two Proportions
Unit 5
AP Statistics

2. Comparing Two Proportions
Comparisons between two percentages are much more common than questions about isolated percentages. And they are more interesting.
We often want to know how two groups differ, whether a treatment is better than a placebo control, or whether this year’s results are better than last year’s.

3. Another Ruler
In order to examine the difference between two proportions, we need another ruler—the standard deviation of the sampling distribution model for the difference between two proportions.
Recall that standard deviations don’t add, but variances do. In fact, the variance of the sum or difference of two independent random quantities is the sum of their individual variances.

4. The Standard Deviation of the Difference Between Two Proportions
Proportions observed in independent random samples are independent. Thus, we can add their variances. So…
The standard deviation of the difference between two sample proportions is
Thus, the standard error is
Remember it’s always a +

5. Assumptions and Conditions
Independence Assumptions:
Randomization Condition: The data in each group should be drawn independently and at random from a homogeneous population or generated by a randomized comparative experiment.
The 10% Condition: If the data are sampled without replacement, the sample should not exceed 10% of the population.
Independent Groups Assumption: The two groups we’re comparing must be independent of each other.

6. Assumptions and Conditions (cont.)
Sample Size Assumption:
Each of the groups must be big enough…
Success/Failure Condition: Both groups are big enough that at least 10 successes and at least 10 failures have been observed in each.

7. The Sampling Distribution
We already know that for large enough samples, each of our proportions has an approximately Normal sampling distribution.
The same is true of their difference.

8. The Sampling Distribution (cont.)
Provided that the sampled values are independent, the samples are independent, and the samples sizes are large enough, the sampling distribution of is modeled by a Normal model with
Mean:
Standard deviation:

9. Two-Proportion z-Interval
When the conditions are met, we are ready to find the confidence interval for the difference of two proportions:
The confidence interval is
where
The critical value z* depends on the particular confidence level, C, that you specify.

10. Steps for two-proportion z-interval
Check Conditions and show that you have checked these!
Randomization Condition: The data in each group should be drawn independently and at random from a homogeneous population or generated by a randomized comparative experiment.
The 10% Condition: If the data are sampled without replacement, the sample should not exceed 10% of the population.
Independent Groups Assumption: The two groups we’re comparing must be independent of each other.
Success/Failure Condition: Both groups are big enough that at least 10 successes and at least 10 failures have been observed in each.
𝒏 𝟏 𝒑 𝟏 ≥𝟏𝟎 𝒏 𝟐 𝒑 𝟐 ≥𝟏𝟎
𝒏 𝟏 𝒒 𝟏 ≥𝟏𝟎 𝒏 𝟐 𝒒 𝟐 ≥𝟏𝟎

11. Steps for TWO PROPORTION Z-interval (cont.)
State the test you are about to conduct
Ex) Two-Proportion z-Interval
Calculate your z-interval
( 𝒑 𝟏 − 𝒑 𝟐 )± 𝒛 ∗ × 𝒑 𝟏 𝒒 𝟏 𝒏 𝟏 + 𝒑 𝟐 𝒒 𝟐 𝒏 𝟐
State your conclusion IN CONTEXT.
We are 95% confident that the support group program could raise the proportion of smokers who manage to quit using the parch by between 2 and 22 percentage points.

12. 2-Proportion Confidence Interval Example
The table below describes the effect of preschool on later use of social services:
Set up a 95% confidences interval. Interpret your results.
Population
Population description
Sample size
Sample proportion 1
Control
𝑛 1 =61
𝑝 1 =0.803 2
Preschool
𝑛 2 =62
𝑝 2 =0.613

13. Everyone into the Pool
The typical hypothesis test for the difference in two proportions is the one of no difference (when they are equal). In symbols, H0: p1 – p2 = 0.
Since we are hypothesizing that there is no difference between the two proportions, that means that the standard deviations for each proportion are the same.
Since this is the case, we combine (pool) the counts to get one overall proportion.

14. Everyone into the Pool (cont.)
The pooled proportion is
where and
If the numbers of successes are not whole numbers, round them first. (This is the only time you should round values in the middle of a calculation.)

15. Everyone into the Pool (cont.)
We then put this pooled value into the formula, substituting it for both sample proportions in the standard error formula:

16. Compared to What?
We’ll reject our null hypothesis if we see a large enough difference in the two proportions.
How can we decide whether the difference we see is large?
Just compare it with its standard deviation.
Unlike previous hypothesis testing situations, the null hypothesis doesn’t provide a standard deviation, so we’ll use a standard error (here, pooled).

17. Two-Proportion z-Test
The conditions for the two-proportion z-test are the same as for the two-proportion z-interval.
We are testing the hypothesis H0: p1 – p2 = 0, or, equivalently, H0: p1 = p2.
Because we hypothesize that the proportions are equal, we pool them to find

18. Steps for two-proportion z-tests
Check Conditions and show that you have checked these!
Randomization Condition: The data in each group should be drawn independently and at random from a homogeneous population or generated by a randomized comparative experiment.
The 10% Condition: If the data are sampled without replacement, the sample should not exceed 10% of the population.
Independent Groups Assumption: The two groups we’re comparing must be independent of each other.
Success/Failure Condition: Both groups are big enough that at least 10 successes and at least 10 failures have been observed in each.
𝒏 𝟏 𝒑 𝒑𝒐𝒐𝒍𝒆𝒅 ≥𝟏𝟎 𝒏 𝟐 𝒑 𝒑𝒐𝒐𝒍𝒆𝒅 ≥𝟏𝟎
𝒏 𝟏 𝒒 𝒑𝒐𝒐𝒍𝒆𝒅 ≥𝟏𝟎 𝒏 𝟐 𝒒 𝒑𝒐𝒐𝒍𝒆𝒅 ≥𝟏𝟎

19. Steps for two-proportion z-tests (cont.)
State the test you are about to conduct
Ex) Two-proportion z-test
Set up your hypotheses
H0:
HA:
Calculate your test statistic
𝒛= 𝒑 𝟏 − 𝒑 𝟐 −𝟎 𝒑 𝒑𝒐𝒐𝒍𝒆𝒅 ∙ 𝒒 𝒑𝒐𝒐𝒍𝒆𝒅 𝒏 𝟏 + 𝒑 𝒑𝒐𝒐𝒍𝒆𝒅 ∙ 𝒒 𝒑𝒐𝒐𝒍𝒆𝒅 𝒏 𝟐
Draw a picture of your desired area under the Normal model, and calculate your P-value.

20. Steps for two-proportion z-tests (cont.)
Make your conclusion.
P-Value
Action
Conclusion
Low
Reject H0
The sample mean is sufficient evidence to conclude HA in context.
High
Fail to reject H0
The sample mean does not provide us with sufficient evidence to conclude HA in context.

21. 2-Proportion z-Test Example
High levels of cholesterol in the blood are associated with higher risk of heart attacks. Will using a drug to lower blood cholesterol reduce heart attacks? The Helsinki Heart Study looked at this question. Middle-aged men were assigned at random to one of two treatments: 2051 men took the drug gemfibrozil to reduce their cholesterol levels, and a control group of 2030 men took a placebo. During the next five years, 56 men in the gemfibrozil group and 84 men in the placebo group had heart attacks. What are the proportions and is the benefit of the drug statistically significant?

22. Calculator Tips
Stat  TESTS
6: 2-PropZTest
Enter values
Calculate

23. What Can Go Wrong?
Don’t use two-sample proportion methods when the samples aren’t independent.
These methods give wrong answers when the independence assumption is violated.
Don’t apply inference methods when there was no randomization.
Our data must come from representative random samples or from a properly randomized experiment.
Don’t interpret a significant difference in proportions causally.
Be careful not to jump to conclusions about causality.

24. Recap
We’ve now looked at inference for the difference in two proportions.
Perhaps the most important thing to remember is that the concepts and interpretations are essentially the same—only the mechanics have changed slightly.

25. Recap (cont.)
Hypothesis tests and confidence intervals for the difference in two proportions are based on Normal models.
Both require us to find the standard error of the difference in two proportions.
We do that by adding the variances of the two sample proportions, assuming our two groups are independent.
When we test a hypothesis that the two proportions are equal, we pool the sample data; for confidence intervals we don’t pool.

26. Assignments: pp. 519 – 522 Day 1: # 1, 7, 9, 18 Day 2: # 3, 10, 20, 22