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 $\sigma^2$.


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

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


  • $1-\alpha$ 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 $t$

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