The standard error of the sample mean is
SEˉx=s√n
The standardized sample mean, or one-sample z statistic,
z=ˉx−μσ/√n
has the N(0, 1) distribution. If the standard deviation σ/√n of ˉx is replaced by the standard error SEˉx=s/√n, the one-sample t statistic
t=ˉx−μs/√n
has the t distribution with n−1 degrees of freedom.
The margin of error for level C confidence is
m=t*×SEˉx=t*s√n
where t* is the value for the t(n−1) density curve with area C between −t* and t*.
A level C confidence interval for the mean μ of a Normal population is
ˉx±m