## Confidence Interval for Mean

Let $X_1, X_2, \cdots , X_{n}$ be a random sample of size $n$ from $N(\mu, \sigma^2)$ with known variance $\sigma^2$.

## Formula

$100(1-\alpha)$% confidence interval for the population mean $\mu$ is

### $\overline{X} - E \leq \mu \leq \overline{X} + E$

where

• $\overline{X} = \dfrac{1}{n} \sum X_i$, is the sample mean,
• $E = Z_{\alpha/2} \dfrac{\sigma}{\sqrt{n}}$ is the margin of error,
• $1-\alpha$ is the confidence coefficient,
• $Z_{\alpha/2}$ is the critical value of $Z$.