Explain a confidence interval without changing its meaning

A frequentist 95% confidence procedure covers the fixed parameter in 95% of repeated samples under its stated model. Once a particular interval has been calculated, that parameter is either inside it or outside it. The 95% describes the procedure's coverage.

A probability statement about the parameter after observing data requires a Bayesian model, including a prior. Neither interpretation should be silently substituted for the other.

An original calculation

Suppose 36 independent observations come from a normal distribution with unknown mean and known standard deviation 12. Their sample mean is five. The standard error of the mean is 12/√36 = 2.

Using the standard normal 97.5th percentile, approximately 1.96, the two-sided 95% interval is:

5 ± 1.96 × 2, or approximately [1.08, 8.92].

The known standard deviation and normal independent observations make the usual normal interval appropriate here. If the standard deviation were estimated from this normal sample, a t procedure would account for that estimation. Other settings need their own justification rather than automatic substitution of 1.96.

What more observations change

Under the same independent model and known standard deviation, quadrupling sample size halves standard error. It does not remove bias, repair a wrong target or make dependent observations independent.

For a simple illustration, suppose observations have common variance σ² and every distinct pair has covariance ρσ². Then:

Var(sample mean) = σ² [1 + (n−1)ρ] / n.

Positive dependence can leave much more uncertainty than the independent formula suggests. This is a specific joint-covariance model, not a universal estimate for any time series.

Selection changes the question

If you inspect many specifications and show only the interval attached to the most favorable estimate, the selection procedure affects interpretation. The usual single-analysis coverage statement does not automatically apply to the selected winner.

Record what was tried, which data selected the procedure and whether a final evaluation remained untouched. A narrow interval after extensive undocumented searching is not automatically strong evidence.

A concise spoken explanation

“Under independent normal observations with known standard deviation, the standard error is two and this procedure gives about [1.08, 8.92]. Its 95% label is repeated-sampling coverage. I would revisit the procedure if the observations were dependent or the specification had been chosen after inspecting these same data.”

The research lab focuses on evaluation design, comparisons and honest interpretation. It is not a comprehensive mathematical-statistics text. For its timing discipline, read the leakage checklist.

The statistics interview preparation page helps distinguish the explanation and evaluation skills practiced in the lab from the additional theory your target role may require.