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 $\mu$ . The construction of CI is based upon the sample mean $\bar{X} =$ and the sample standard deviation from data of sample size . The most commonly used confidence interval is a two-sided CI which is centered at the mean $\bar{X}$ and extends either side an equal amount.

$\displaystyle\bigg(\bar{X} - t_{\alpha/2,n-1}\frac{S}{\sqrt{n}} =$ , $\displaystyle\bar{X} + t_{\alpha/2,n-1}\frac{S}{\sqrt{n}} =$ $\displaystyle\bigg)$

If the standard deviation $\sigma =$ is known, the critical point $t_{\alpha/2,n-1}$ should be replaced by $z_{\alpha/2}$ of the standard normal distribution.

$\displaystyle\bigg(\bar{X} - z_{\alpha/2}\frac{\sigma}{\sqrt{n}} =$ , $\displaystyle\bar{X} + z_{\alpha/2}\frac{\sigma}{\sqrt{n}} =$ $\displaystyle\bigg)$