e-Statistics

Confidence Interval

A confidence interval (CI) provides a range of plausible values for the unknown population mean $\mu$ . The choice of the confidence level $(1-\alpha) =$ is typically 90%, 95% or 99%, and represents the chance that the CI does indeed contain the true population mean $\mu$ . It is usually associated with significance level $\alpha$ . The construction of CI is based upon the sample mean $\bar{X}$ = and the sample standard deviation = from data of sample size n = . The most commonly used confidence interval is a two-sided CI which is centered at the mean $\bar{X}$ .

$\bar{X} \pm t_{\alpha/2,n-1}\dfrac{S}{\sqrt{n}}$ = ( , )

CI extends either side an equal amount, and the amount $t_{\alpha/2,n-1}\dfrac{S}{\sqrt{n}}$ = is called the margin of error.

When a precise estimate of standard deviation $\sigma$ = is known (and often assumed for a sufficiently large ), $t_{\alpha/2,n-1}$ of t-distribution should be replaced by the critical value $z_{\alpha/2}$ of the standard normal distribution.

$\bar{X} \pm z_{\alpha/2}\dfrac{\sigma}{\sqrt{n}}$ = ( , )