Where:

• $\hat p$ is the sample proportion
• $z_{\alpha/2}$ is the critical value from the standard normal distribution
• $n$ is the sample size

• When both np ≥ 10 and n(1-p) ≥ 10
• Good for quick calculations and teaching purposes

• Can produce intervals outside [0,1]
• Poor coverage for small samples or extreme proportions
• Symmetric intervals may not reflect the true uncertainty

In a sample of 200 people, 30 are left-handed (p̂ = 0.15). For 95% confidence:

Where $z_{\alpha/2}$ is the same critical value used in the normal approximation method.

• Always produces intervals within [0,1]
• Better coverage properties than normal approximation
• Works well for small samples and extreme proportions
• Accounts for the uncertainty in the variance estimate

• Recommended for most practical situations
• Especially good when normal approximation assumptions aren't met
• Default choice for modern statistical software

For x successes in n trials:

Where Beta represents the inverse beta cumulative distribution function.

• Guarantees exact coverage probability
• No approximations involved
• Works for any sample size
• Always produces valid intervals [0,1]

• Conservative (actual coverage often exceeds nominal level)
• Can produce unnecessarily wide intervals
• Computationally more complex

• When you need guaranteed coverage
• For regulatory or safety-critical applications
• When sample sizes are very small

• n < 30: Use Wilson Score or Clopper-Pearson
• 30 ≤ n < 100: Wilson Score recommended, Normal Approximation acceptable if assumptions met
• n ≥ 100: All methods generally work well, Wilson Score still preferred