Essentials of Mathematical Probability and StatisticsA First Course For the Mathematically Inclined

Section 10.5 Interval Estimates - Confidence Interval for $\mu$

As with the confidence intervals above for proportions, the Central Limit Theorem also allows you to create an interval centered on a sample mean for estimating the population mean $\mu\text{.}$

Once again, utilize the Central Limit Theorem. Notice that the symmetrical confidence interval

is equivalent to

in which the middle term can be approximated using a standard normal variable and therefore this statement is approximately

Using the symmetry of the standard normal distribution about Z=0 gives

and so to determine E again requires the inverse of the standard normal distribution function. Using an appropriate $z_{\alpha /2}$ (as determine in a manner described in the previous section) gives a confidence interval for the mean

with confidence level $1-\alpha$ and margin of error

It should be noted that the use of the Central Limit Theorem makes the use of InvNorm an approximation. It can be shown that so long as n is larger than 30 then generally this approximation is reasonable.

Additionally, this derivation assumes that $\mu$ is not known...indeed the goal is to approximate that mean using $\overline{x}\text{...}$but that $\sigma$ is known. This is often not the case. It can however be shown that if n is larger than 30, replacing $\sigma$ with the sample standard deviation s gives an acceptable confidence interval.

Notice, in practice you might want to take n to be a little larger than the absolute minimum value prescribed above since you are dealing with approximations (Central Limit Theorem and the use of an estimate for s rather than the actual $\sigma\text{.}$)