Estimating a Difference in Proportions
Learning Targets
- Determine conditions for confidence interval for a difference between two proportions.
- Construct and interpret a confidence interval for a difference between two proportions.
Key Concepts
- Confidence Interval: A way to estimate the difference in proportions between two populations (p₁ and p₂).
- Difference Calculation: The statistic used is p₁ - p₂, where p₁ and p₂ are population proportions.
Conditions for Constructing Confidence Intervals
- Random: Data comes from independent random samples or randomized experiments.
- 10% Condition: When sampling without replacement, n₁ < 0.10N₁ and n₂ < 0.10N₂.
- Large Counts: All counts of successes and failures must be at least 10 (n₁p₁, n₁(1 - p₁), n₂p₂, n₂(1 - p₂) ≥ 10).
- For a C% confidence interval for p₁ - p₂:
(p₁ - p₂)
ightarrow ext{statistic} \pm z^* \cdot SE{p1 - p2}
Where, SE{p1 - p2} = \sqrt{ \frac{p1(1-p1)}{n1} + \frac{p2(1-p2)}{n2} }
- z* is the critical value from the standard Normal distribution.
Interpretation of Confidence Interval
- If the confidence interval excludes 0, it suggests a significant difference in population proportions. If it includes 0, no significant difference is found.
Example Application: Brand Names Preference
- Survey showed:
- U.S. proportion (p₁): 0.26
- German proportion (p₂): 0.22
- Confidence interval calculated to be (0.01, 0.07), indicating we are 95% confident that the true difference in proportions lies within this interval.
Conclusion Guidelines
- If 0 is not included in the confidence interval, significant evidence exists for a difference; if it is included, no claim can be confidently made about the difference.