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 $t$