## large sample size For large samples with a [[normal distribution]], including those approximately normal by the [[Central Limit Theorem]], a confidence interval is constructed from the [[standard normal distribution]]. An approximate $100(1-\alpha)\%$ confidence interval for the mean $\mu$ is given by $\bar X \pm z_{\alpha/2} \sqrt{\frac{s^2}{\sqrt{n}}}$ where $s^2$ is the [[sample variance]] and $z_{\alpha/2}$ is the [[critical value]] from the standard normal distribution. ## small sample size For small samples, if and only if the population has the normal distribution, the confidence interval is constructed from the [[t-distribution]]. An approximate $100(1-\alpha)\%$ confidence interval for the mean $\mu$ is given by $\bar X \pm t_{\alpha/2, n-1} \sqrt{\frac{s^2}{\sqrt{n}}}$ where $s^2$ is the [[sample variance]] and $t_{\alpha/2, n-1}$ is the [[critical value]] from the t-distribution.