## Confidence Interval for the mean ($\sigma$ unknown)

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

## Formula

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

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

where,

• $1-\alpha$ is the confidence coefficient,
• $\overline{X} = \dfrac{1}{n}\sum_{i=1}^n X_i$ is the sample mean,
• $E = t_{(\alpha/2,n-1)} \frac{s}{\sqrt{n}}$is the margin of error,
• $s =\sqrt{\frac{1}{n-1}\sum (X_i - \overline{X})^2}$ is the sample standard deviation
• $t_{(\alpha/2,n-1)}$ is the critical value of $t$