## 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 $t$