Interval estimation for variances

Examples

$X_{1}, \dots, X_{n} \sim N (μ, σ^{2})$ , $μ$ is known: use Sampling Distributions of $χ_{n}^{2}$

\begin{aligned} P (σ_{l}^{2} \leq σ^{2} \leq σ_{u}^{2}) = 1 - α \\ P (\sum \frac{(x_{i} - μ)^{2}}{σ_{u}^{2}} \leq \frac{(x_{i} - μ)^{2}}{σ_{u}^{2}} \leq \frac{(x_{i} - μ)^{2}}{σ_{l}^{2}}) \end{aligned}

If $μ$ is unknown

\begin{aligned} \dots \\ C I for σ^{2} : (\frac{\sum (x_{i} - \bar{x})^{2}}{χ_{1 - \frac{α}{2}, n - 1}^{2}}, \frac{\sum (x_{i} - \bar{x})^{2}}{χ_{\frac{α}{2}, n - 1}^{2}}) = (\frac{(n - 1) s^{2}}{χ_{1 - \frac{α}{2}, n - 1}^{2}}, \frac{(n - 1) s^{2}}{χ_{\frac{α}{2}, n - 1}^{2}}) \end{aligned}