Confidence Interval for the Mean (z)

Confidence interval for the mean (z)

If you already know the population standard deviation σ, or your sample is large enough that n > 30 and the Central Limit Theorem kicks in, the confidence interval for the mean comes out to IC = x̄ ± z · σ/√n. Which critical value you plug in depends on the level you want: z = 1.645 at 90%, z = 1.96 at 95%, z = 2.576 at 99%. That σ/√n piece is the standard error, and it falls off with √n. So if you want to cut the margin in half, you need four times the sample. For small samples where σ is a mystery (n < 30), switch to Student's t. Worked example: x̄ = 50, σ = 8, n = 36, 95% → IC = 50 ± 1.96 · 8/6 = [47.39, 52.61].

Applications

You'll see it behind the margin of error quoted in election polls, in clinical trials, in statistical quality control, in A/B testing, when labs validate an analytical method, and when economic indicators like IBGE or IPCA figures get estimated.

FAQ

Does the CI mean μ has 95% chance of being inside? No, and this trips almost everyone up. The right reading is frequentist. Repeat the sampling 100 times and roughly 95 of the intervals you build this way will contain the true μ. But for the one interval in front of you, μ is in it or it isn't. There's no probability left.

z or t? Reach for z when σ is known or n > 30. Reach for t when σ is unknown and the sample is small. The t-Student distribution carries heavier tails than z, which is exactly what offsets the extra uncertainty you take on by estimating σ from s.

How to reduce the margin of error? Collect more data, since the margin falls with √n. You can also try to trim the variable's own variability, or drop your confidence level. Going from 95% to 90% does tighten the interval, but it also raises your odds of missing μ.