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I have a $X_1, \cdots , X_n$ random samples from a $N(\mu,\sigma^2)$. $\mu$ is known, but $\sigma^2$ is unknown. I would like to know how to go about constructing a $(1-\alpha)$100% shortest confidence interval for $\sigma^2$.

I know how how to construct that of a known variance, but unknown mean. I'm bit confused as how to proceed when the situation is reversed.

Thank you.

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    The shortest interval can be found by solving for $a$ and $b$ such that $a^2 f(a) = b^2 f(b)$ subject to $\int_{a}^b f(x) \,\mathrm{d}x = \alpha$ where $f$ is the density of a chi-square distribution with $n$ degrees of freedom. No closed form exists, so people in practice tend to use an *equal-tails* confidence interval, which is the one that @Sivaram gave. However, the equal-tails interval is **not** the shortest possible.2011-06-23
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    The integral in my previous comment should obviously be equal to $1-\alpha$. Apologies for the typo.2011-06-23

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