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 $ S =$ 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}$ 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)$