Confidence Interval Formulas
For Mean (unknown σ): CI = x̄ ± t × (s/√n)
For Proportion: CI = p̂ ± z × √(p̂(1−p̂)/n)
Where x̄ = sample mean, z = critical z-value, σ = population SD, s = sample SD, n = sample size, p̂ = sample proportion.
Common Critical Values
| Confidence Level | z* (two-tailed) | α |
|---|---|---|
| 90% | 1.645 | 0.10 |
| 95% | 1.960 | 0.05 |
| 99% | 2.576 | 0.01 |
| 99.9% | 3.291 | 0.001 |
What Does "95% Confidence" Mean?
A 95% confidence interval means: if you repeated this study 100 times, approximately 95 of the 100 confidence intervals would contain the true population parameter. It does not mean there is a 95% probability the parameter is in this specific interval — once calculated, the interval either contains the parameter or it doesn't.
Frequently Asked Questions
When should I use z vs t critical values?
Use z when the population standard deviation (σ) is known (rare in practice) or when n > 30. Use t when σ is unknown and you only have the sample standard deviation s — which is the typical situation. With large samples (n > 30), z and t give nearly identical results.
How does sample size affect the confidence interval?
Larger samples produce narrower (more precise) confidence intervals. The margin of error is proportional to 1/√n — so to halve the margin of error, you need to quadruple the sample size. This is why large samples are so valuable in research.
What is the margin of error?
The margin of error (E) is the ± value added and subtracted from the sample statistic to create the interval: E = z × (σ/√n) for means. The confidence interval is then [x̄ − E, x̄ + E]. In polling, a "±3% margin of error" typically refers to a 95% CI on a proportion.