Confidence Intervals

Definition

A $100(1-\alpha)\%$ confidence interval for parameter $\theta$ is a random interval $(L, U)$ such that:

$P(L < \theta < U) = 1 - \alpha$

Interpretation

If we repeated the sampling procedure many times, about $100(1-\alpha)\%$ of the computed intervals would contain the true parameter.

Not: There is a $(1-\alpha)$ probability that $\theta$ lies in this interval. ($\theta$ is fixed, the interval is random.)

CI for Mean (Known Variance)

$X_1, \ldots, X_n$ from population with known $\sigma^2$:

$\bar{X} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$

where $z_{\alpha/2} = \Phi^{-1}(1 - \alpha/2)$.

Derivation: By CLT, $\frac{\bar{X} - \mu}{\sigma/\sqrt{n}} \approx N(0,1)$.

$P\left(-z_{\alpha/2} < \frac{\bar{X} - \mu}{\sigma/\sqrt{n}} < z_{\alpha/2}\right) \approx 1 - \alpha$

CI for Mean (Unknown Variance)

Use $t$-distribution with $n-1$ degrees of freedom:

$\bar{X} \pm t_{n-1, \alpha/2} \frac{S}{\sqrt{n}}$

For large $n$, $t_{n-1} \approx z$.

Common Confidence Levels

Confidence Level	$\alpha$	$z_{\alpha/2}$
90%	0.10	1.645
95%	0.05	1.96
99%	0.01	2.576

Example: Heights

$n = 25$ heights with $\bar{X} = 170$ cm, $S = 10$ cm.

95% CI: $170 \pm 1.96 \times \frac{10}{\sqrt{25}} = 170 \pm 3.92$

95% CI: $(166.08, 173.92)$ cm.

Interpretation

We are 95% confident that the true mean height lies between 166.08 cm and 173.92 cm.

“Confident” refers to the procedure, not the probability of this specific interval.

Summary

Parameter	CI Formula
Mean ($\sigma$ known)	$\bar{X} \pm z_{\alpha/2}\sigma/\sqrt{n}$
Mean ($\sigma$ unknown)	$\bar{X} \pm t_{n-1, \alpha/2}S/\sqrt{n}$