Confidence Intervals, explained.
A frequentist confidence interval is produced by a procedure with a specified long-run coverage rate under its assumptions.
Why it happens
Sample means are Normal with true mean zero and known population standard deviation one. Each interval is mean ± 1.96 divided by the square root of sample size; forty intervals show realised coverage.
A frequentist confidence interval is produced by a procedure with a specified long-run coverage rate under its assumptions.
Read the result
Count intervals containing zero after the reveal. Coverage is a property of the repeated procedure, not a posterior probability for one fixed interval.
A worked example
Forty studies of a fixed mean
Generate forty independent Normal sample means with the selected sample size and known variance.
Reveal the fixed generating mean of zero and inspect which intervals cross its vertical reference line.
Increase sample size to narrow intervals while keeping the same nominal long-run coverage rule.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
Sample means are Normal with true mean zero and known population standard deviation one. Each interval is mean ± 1.96 divided by the square root of sample size; forty intervals show realised coverage.
Where this idea is useful
A practical use
Report the estimate and its uncertainty together, and ask which sampling and model assumptions support the stated coverage.
A common misconception
“A 95% interval contains 95% of observations.”
The interval estimates a population parameter. It is not a prediction interval for individual observations or a description of their central spread.
What this explanation leaves out
- Coverage depends on independent Normal sampling and known variance. A fixed interval has no frequentist probability that its fixed parameter lies within it.
Must exactly thirty-eight intervals contain zero?
No. Ninety-five percent is a long-run procedure rate under the assumptions; a batch of forty can have a different realised count.
Which parameter, sampling process and interval procedure justify your uncertainty statement?
Associated thinkers
Associations marked provisional are awaiting source review.
Further reading
Explore the original research or the teaching reference behind this experiment.