Example 1: Confidence Interval for population mean

The sampling distribution for the sample mean tends in the limit of large $n$ to $\overline{X}\sim N(\mu,\sigma^2/n)$. Therefore, the pivotal quantity or test statistic $Z=(\overline{X}-\mu)/(\sigma/\sqrt{n})$ is distributed as $N(0,1)$. The $(1-\alpha)100\%$ confidence interval for $\mu$ can be written

$$\overline{x}-Z_c\frac{\sigma}{\sqrt{n}}\leq \mu \leq \overline{x}+Z_c\frac{\sigma}{\sqrt{n}}$$ (5.1)

where $Z_c(\alpha)=-\Phi^{-1}(\alpha/2)$ is the the half-width of the $(1-\alpha)100\%$ confidence interval measured in standard errors. $Z_c$ is sometimes referred to as a critical value. Table 4.1 gives some critical values for various confidence limits:

Table: Critical values for various common confidence levels

$\alpha$	$1-\alpha$	$Z_c$	Description
0.50	0.50	0.68	50% C.I. $\pm$ one ``probable error''
0.32	0.68	1.00	68% C.I. $\pm$ one ``standard error''
0.10	0.90	1.65	90% C.I.
0.05	0.95	1.96	95% C.I. about $\pm$ two standard errors
0.001	0.999	3.29	99.9% C.I. about $\pm$ three standard errors