A Confidence Interval (CI) provides a range of plausible values for an unknown population parameter (like the mean $\mu$ or proportion $p$). It is constructed from sample data. For example, a “95% Confidence Interval” means that if we took many samples and built a CI from each, 95% of them would contain the true population parameter.

Calculating for Means

For the population mean $\mu$, the formula depends on the sample size ($n$) and whether the population standard deviation ($\sigma$) is known.
Large Sample ($n \ge 30$): We use the Z-statistic (Standard Normal Distribution). $$ \bar{x} \pm Z_{\alpha/2} \left( \frac{s}{\sqrt{n}} \right) $$
Small Sample ($n < 30$): We use the T-statistic (Student’s t-Distribution), which has “heavier tails” to account for the uncertainty in estimating the standard deviation. $$ \bar{x} \pm t_{\alpha/2, df} \left( \frac{s}{\sqrt{n}} \right) $$

Calculating for Proportions

For population proportion $p$, we typically use the Z-statistic approximation, provided the sample size is large enough ($np \ge 10$ and $n(1-p) \ge 10$). $$ \hat{p} \pm Z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $$