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Justifying a Claim Based on a Confidence Interval for a Difference of Population Proportions
When comparing two population proportions, a confidence interval for their difference provides a range of plausible values for $ (p_1 - p_2) $ . This interval is a powerful tool for making an informed decision or justifying a claim about whether there’s a significant difference between the two proportions, or if one proportion is greater/less than the other.
The Purpose of Justification
The goal is to determine if the confidence interval for $ (p_1 - p_2) $ supports a specific claim about the relationship between $ p_1 $ and $ p_2 $ . Common claims include:
- $ p_1 \neq p_2 $ (there is a difference)
- $ p_1 > p_2 $ (or $ p_1 - p_2 > 0 $ )
- $ p_1 < p_2 $ (or $ p_1 - p_2 < 0 $ )
- $ p_1 = p_2 $ (there is no difference)
For a deeper understanding of how these intervals are constructed, refer to Constructing a Confidence Interval for a Population Proportion and Sampling Distributions for Differences in Sample Proportions.
Interpreting the Confidence Interval
The key to justifying a claim lies in observing where the value 0 falls relative to the confidence interval $ (L, U) $ , where $ L $ is the lower bound and $ U $ is the upper bound.
Case 1: The Interval Contains 0
If the confidence interval $ (L, U) $ for $ (p_1 - p_2) $ includes 0:
- This means that 0 is a plausible value for the true difference between $ p_1 $ and $ p_2 $ .
- Therefore, we do not have convincing evidence to claim that $ p_1 $ and $ p_2 $ are different.
- It is plausible that $ p_1 = p_2 $ .
- We cannot conclude that one proportion is definitively greater or less than the other.
Example: A 95% confidence interval for $ (p_1 - p_2) $ is $ (-0.05, 0.03) $ . Since this interval contains 0, we cannot conclude there’s a significant difference between $ p_1 $ and $ p_2 $ at the 95% confidence level.
Case 2: The Interval Does Not Contain 0
If the confidence interval $ (L, U) $ for $ (p_1 - p_2) $ does not include 0, this provides convincing evidence of a difference between $ p_1 $ and $ p_2 $ . There are two sub-cases:
-
Interval is Entirely Above 0 ( $ L > 0 $ ):
- This means all plausible values for $ (p_1 - p_2) $ are positive.
- Therefore, we have convincing evidence that $ p_1 - p_2 > 0 $ , which implies $ p_1 > p_2 $ .
- We can claim that $ p_1 $ is significantly greater than $ p_2 $ .
Example: A 95% confidence interval for $ (p_1 - p_2) $ is $ (0.02, 0.08) $ . Since the entire interval is above 0, we are 95% confident that $ p_1 $ is between 2% and 8% higher than $ p_2 $ . We have convincing evidence that $ p_1 > p_2 $ .
-
Interval is Entirely Below 0 ( $ U < 0 $ ):
- This means all plausible values for $ (p_1 - p_2) $ are negative.
- Therefore, we have convincing evidence that $ p_1 - p_2 < 0 $ , which implies $ p_1 < p_2 $ .
- We can claim that $ p_1 $ is significantly less than $ p_2 $ .
Example: A 95% confidence interval for $ (p_1 - p_2) $ is $ (-0.10, -0.03) $ . Since the entire interval is below 0, we are 95% confident that $ p_1 $ is between 3% and 10% lower than $ p_2 $ . We have convincing evidence that $ p_1 < p_2 $ .
Summary Table for Justifying Claims
| Claim about $ p_1 $ vs. $ p_2 $ | Condition on Interval $ (L, U) $ for $ (p_1 - p_2) $ | Conclusion |
|---|---|---|
| $ p_1 = p_2 $ | Interval contains 0 | We do not have convincing evidence at the given confidence level to conclude that $ p_1 $ and $ p_2 $ are different. It is plausible that $ p_1 = p_2 $ . |
| $ p_1 \neq p_2 $ | Interval does not contain 0 ( $ L > 0 $ or $ U < 0 $ ) | We have convincing evidence at the given confidence level that $ p_1 $ and $ p_2 $ are different. (Specify if $ p_1 > p_2 $ or $ p_1 < p_2 $ based on whether the interval is entirely positive or entirely negative). |
| $ p_1 > p_2 $ | Interval is entirely above 0 ( $ L > 0 $ ) | We have convincing evidence at the given confidence level that $ p_1 > p_2 $ , because all plausible values for $ (p_1 - p_2) $ are positive. |
| $ p_1 < p_2 $ | Interval is entirely below 0 ( $ U < 0 $ ) | We have convincing evidence at the given confidence level that $ p_1 < p_2 $ , because all plausible values for $ (p_1 - p_2) $ are negative. |
The Role of Confidence Level
The confidence level (e.g., 90%, 95%, 99%) reflects the strength of our conclusion. A 95% confidence interval implies that if we were to repeat the sampling process many times, 95% of the intervals constructed would capture the true difference.
A higher confidence level will result in a wider interval, making it more likely to contain 0, and thus less likely to show a significant difference. Conversely, a lower confidence level will result in a narrower interval, making it more likely to exclude 0.
Confidence Intervals vs. Hypothesis Tests
Justifying a claim based on a confidence interval is closely related to performing a Setting Up a Test for the Difference of Two Population Proportions. In fact, a confidence interval can be used to perform a two-sided hypothesis test:
- If a $ (1-\alpha) \times 100% $ confidence interval for $ (p_1 - p_2) $ contains 0, then a two-sided hypothesis test at significance level $ \alpha $ would fail to reject the null hypothesis $ H_0: p_1 = p_2 $ .
- If the interval does not contain 0, then the two-sided test would reject $ H_0 $ .
Confidence intervals provide more information than a hypothesis test alone because they give a range of plausible values for the true difference, not just a “yes/no” decision.
The general form of a confidence interval for the difference of two population proportions is: $$ (\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}} $$ where $ \hat{p}_1 $ and $ \hat{p}_2 $ are the sample proportions, $ n_1 $ and $ n_2 $ are the sample sizes, and $ z^* $ is the critical value from the standard normal distribution corresponding to the desired confidence level.