Confidence Intervals for Proportions — Estimating p
Foundations of Statistics
Measuring Uncertainty in Binary Outcomes
Confidence intervals for proportions quantify uncertainty in estimated percentages, from conversion rates to disease prevalence. Different interval methods handle edge cases and small samples with varying accuracy.
- Political Polling — Reporting election results with appropriate margins of error
- Marketing — Estimating true conversion rates from A/B test samples
- Public Health — Tracking disease prevalence with quantified uncertainty
Proportion intervals are essential whenever outcomes are yes/no or success/failure.
Core Concepts
Confidence intervals for proportions estimate the true population proportion from sample data. The Wald interval is the most common, but the Wilson interval performs better for extreme proportions.
Derivation of the Wald Interval
Wilson Score Interval
Sample Size for Desired Margin of Error
Worked Example: Wald Interval
A poll of voters finds support for a candidate. Construct a 95% CI.
Step 1. Check conditions: and . ✓
Step 2. Standard error: .
Step 3. Margin of error: .
Step 4. The 95% Wald CI: .
Interpretation: We are 95% confident that between 53.7% and 62.3% of voters support the candidate.
Worked Example: Wilson Interval
For the same data (, ), compute the Wilson interval.
Step 1. Compute the components: , .
Step 2. The center: .
Step 3. The standard error term: .
Step 4. The Wilson CI: .
Lower: . Upper: .
The Wilson CI is — very close to the Wald interval here because is large and is not extreme.