What is a Confidence Interval?
A Confidence Interval (CI) gives us a range of values within which we believe the true value of a population parameter (like mean or proportion) lies, with a certain level of confidence (like 95%).
Example: "We are 95% confident that the population mean lies between 65 and 75."
General Formula
For population mean μ (when σ is known):
CI = X̄ ± Zα/2 × (σ / √n)
For proportion p:
CI = p̂ ± Zα/2 × √[p̂(1 - p̂) / n]
Where:
- X̄ = sample mean
- p̂ = sample proportion
- Zα/2 = Z-score for desired confidence (e.g., 1.96 for 95%)
- σ = population standard deviation
- n = sample size
Interpreting Confidence Intervals
A 95% CI does not mean there's a 95% chance the population mean is in the interval. It means that **if we repeated the sampling process many times**, 95% of the resulting intervals would contain the true population mean.
Factors Affecting Width of CI
- Sample Size (n): Larger sample → narrower interval
- Confidence Level: Higher confidence (e.g., 99%) → wider interval
- Standard Deviation: Greater variability → wider interval
Example
Suppose X̄ = 70, σ = 10, n = 100, and confidence level = 95%.
Z0.025 = 1.96
CI = 70 ± 1.96 × (10 / √100) = 70 ± 1.96 = [68.04, 71.96]