Confidence Interval for Proportion Calculator

Understanding Confidence Intervals for Proportions

A confidence interval for a proportion estimates the range within which the true population proportion is likely to fall, based on sample data and the chosen confidence level.

Formula for Confidence Interval of Proportion

The formula for calculating the confidence interval for a sample proportion \( \hat{p} \) is:

$$ CI = \hat{p} \pm Z \cdot \sqrt{\frac{\hat{p} (1 - \hat{p})}{n}} $$

where:

• \( \hat{p} \): Sample proportion
• \( n \): Sample size
• \( Z \): Z-score corresponding to the desired confidence level (e.g., 1.96 for 95%)

Real-Life Example: Estimating Voting Preferences

Suppose a poll found that 52% of a sample of 1,000 voters support a particular candidate. To determine the 95% confidence interval for this proportion:

• Step 1: Set \( \hat{p} = 0.52 \) and \( n = 1000 \).
• Step 2: Use a 95% confidence level, so \( Z = 1.96 \).
• Step 3: Calculate the standard error:

$$ SE = \sqrt{\frac{0.52 \times (1 - 0.52)}{1000}} \approx 0.0157 $$

• Step 4: Calculate the margin of error:

$$ ME = 1.96 \times 0.0157 \approx 0.0308 $$

• Step 5: Determine the confidence interval:

$$ CI = 0.52 \pm 0.0308 = [0.4892, 0.5508] $$

This means we can be 95% confident that between 48.92% and 55.08% of the population supports the candidate.