Unknown population standard deviation

If we know the value of the population standard deviation, σ, an interval estimate of the form

has a confidence level of 0.95 — i.e. it is a 95% confidence interval.

In practice however, the value of σ is rarely known.

It is tempting to simply replace σ in the formula by its sample equivalent, s.

However replacing σ with s makes the confidence interval more variable and this means that it is less likely include µ — the confidence level is less than 95%. If the sample size, n, is large, the confidence level is close to 95%, but with smaller sample sizes the true confidence level can be much less than the target 95%. For example,

The confidence interval must be modified if σ is unknown.