Confidence Intervals

A confidence interval is a random interval constructed from the sample; for a normal mean the classic form is

$CI = \bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$,

with $z_{\alpha/2}$ the standard-normal quantile and $\sigma/\sqrt{n}$ the standard error — width is set by both precision and confidence level. The frequentist interpretation is coverage: across repeated sampling, $100(1-\alpha)\%$ of intervals contain the true value; for a single interval, the notion of a “probability the parameter falls inside” does not apply — the parameter is fixed, the interval is random.

quantifying estimation uncertainty, reporting error bars for A/B experiments and model evaluation; interviews typically probe the strict coverage interpretation, switching to the t-distribution when $\sigma$ is unknown, and the duality with two-sided tests.

a point estimate gives a single number with no precision information; a CI reports both the estimate and its uncertainty: raising confidence from 95% to 99% ($z_{\alpha/2}$ from 1.96 to 2.576) or halving $n$ widens the interval substantially; by duality, a 95% CI that excludes $\mu_0$ is equivalent to rejecting $H_0: \mu = \mu_0$ at $\alpha = 0.05$; when no closed form exists (complex statistics), the Bootstrap supplies approximate intervals.

The 95% CI $\bar{x} \pm z_{\alpha/2}\sigma/\sqrt{n}$ for a normal mean with known variance: meaning of each component?

Strict coverage interpretation: why can't we say the true value lies in the interval with 95% probability?

How does the width change when confidence goes 95%→99%, or when $n$ doubles?

Why the t-distribution when $\sigma$ is unknown? Difference from the normal approximation?

What is the duality between confidence intervals and two-sided hypothesis tests?