1-Variance $\chi^2$ Test - Odd behavior

by Jared Goguen   Last Updated April 18, 2018 19:19 PM

Every resource I have found states that the sample variance of a normal random variable follows a $\chi^2$-distribution such that $(n-1)\dfrac{s^2}{\sigma^2} \sim \chi^2_{n-1}$.

I am confused about how to find the $P$-value for a two-tailed test when the $s \approx \sigma$. Suppose $s$ is very slightly less than $\sigma$. I would expect that the $P$-value of a two-tailed test could be found by multiplying the left-tail area by $2$. However, this produces a $P$-value that is greater than $1$.

Many technologies instead use the right-tail area (which is less than $0.5$) and multiply that by $2$, but this does not sit well with me because (a) the sample variance is less than the population variance, and (b) this makes the upper limit for a $P$-value strictly less than $1$.

Am I missing something here? Is there a good way to calculate the $P$-value for this type of test that does not run into these issues?

example distribution

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