An interval with a coverage promise
A confidence interval turns a point estimate into a range with a stated success rate: a 95% interval is built by a method that captures the true parameter in 95% of repeated samples. The confidence belongs to the method, not to any single interval: this particular range either contains the truth or it does not, and the 95% describes the procedure's long-run record.
The bakery sample: 8 loaves, mean 44.5, s = 4.75, standard error 4.75 / √8 = 1.68. With t(0.025, 7) = 2.365, the margin is 2.365 × 1.68 = 3.97, and the 95% interval runs from 40.53 to 48.47. Note what the interval does that the test did not: it shows the claim of 40 sitting just outside the range, and it shows the plausible means extending to 48.5: evidence about size, not only about existence.
What widens and narrows an interval
Sample size narrows it (standard error falls like 1/√n: quadruple n to halve the width). Higher confidence widens it (99% costs more width than 95%, for the same data). Noisier data widen it. There is no fourth lever: anyone promising a narrow interval at high confidence from small noisy data is selling something.
Two standard misreadings. (1) "There is a 95% probability the mean is in this interval": in the frequentist frame the mean is fixed and the interval is the random part; the correct statement is about the method's capture rate. (2) "95% of the data lie in this interval": the interval brackets the mean, which is far narrower than the spread of individual values; the data's own range here is 38 to 51, wider than the interval for the mean.
Intervals and tests say the same thing two ways
A 95% interval that excludes the null value corresponds exactly to rejecting that null at the two-tailed 5% level. The interval is the richer report: every value inside it is a null that would not be rejected, so the reader sees the full set of plausible parameters at a glance.
Frequently asked questions
What does 95% confident actually mean?
That the procedure captures the true parameter in 95% of repeated samples. For your single interval, the parameter is either inside or not; the 95% is the method's track record, which is what justifies trusting it.
Why is my interval so wide?
Small sample, variable data, or high confidence requested. Width scales with s over root n times the critical value. The only real cure is more data or less noise; choosing 90% instead of 95% buys width at the price of confidence.
Should I report 95% or 99%?
95% is the working standard in most fields. Use 99% when a false claim is expensive or when your field's convention demands it, and state the level either way; an interval without its level is half a result.
Can a confidence interval include impossible values?
Yes, near boundaries: a proportion interval computed by the simple Wald method can dip below 0. Better methods (Wilson) respect the bounds. Treat out-of-range endpoints as a sign to switch methods, not as findings.