Confidence Intervals

A confidence interval turns a single estimate into an honest range: instead of "the mean is 495g," it says "we’re 95% confident the true mean lies between 491 and 499g."

Where 1.96 comes from

Straight from the z table: 95% of a normal distribution lies between −1.96 and +1.96 (each tail holds 2.5%, and P(Z < 1.96) = 0.97500). The famous constant is just a table lookup that everyone memorized.

Worked example

Sample of n = 25 bags, mean 495g, σ = 10: the 95% interval is 495 ± 1.96 × (10/√25) = 495 ± 3.92 → (491.1, 498.9). Note what it excludes: 500. That’s the same verdict the z-test gave via its p-value — intervals and tests are two views of one calculation.

How to read it correctly

The 95% refers to the procedure: if you repeated the sampling many times, ~95% of the intervals built this way would capture the true mean. Any single interval either contains it or doesn’t — the confidence lives in the method. Wider intervals come from smaller samples, larger σ, or higher confidence levels; the √n in the denominator is why quadrupling the sample only halves the width.