Chi-squared confidence interval for variance When constructing, for example, a $90\%$ confidence interval for the population variance using the chi-squared distribution, we have:
\begin{align}
& P\left(a<\frac{(n-1)S^2}{\sigma^2}<b\right) \\
= {} & P\left(\frac{\sum(\bar{X}-X_i)^2}{b}<\sigma^2<\frac{\sum(\bar{X}-X_i)^2}{a}\right)=0.9, \\ & \text{ where } \frac{(n-1)S^2}{\sigma^2}\sim\chi_{n-1}^2.
\end{align}
In my course, we then find $a, b$ such that $$P(\chi_{n-1}^{2}<a)=P(\chi_{n-1}^{2}>b)=0.05.$$
My question is, given that the chi-squared distribution is asymmetric for small $n,$ why do we pick $0.05$ for both sides? Surely we’d have a shorter confidence interval if we weighted more of the $0.1$ probability to one of the sides, instead of splitting equally?

One answer is why not use equal probabilities for both tails? Suppose we do not, we then produce asymmetric confidence intervals in terms of probability. Let us take an extreme example, to make a 95% confidence interval, we would usually set the interval at 2.5% lower, and 2.5% upper tail. Now we decide we want an extreme example, so we choose 0% left tail and a 5% upper-tail of a heavy right tailed distribution. What is that? It is a 5% one-tailed (right) answer to the question of what the probability is of an $X_i\geq$ the larger confidence interval upper bound. So, why balance the probabilities? So that the probability of an answer lower than the confidence interval is the same the probability of an answer higher than the confidence interval, which then gives us a balanced or two-tailed answer, which is the probability of a $X_i\neq$ a value within the confidence interval.
It is not unreasonable to ask if there are situations in which one wants unequal probabilities for the tails. Here is an arbitrary example for which the probabilities are not the final answers sought. Suppose that the x-axis measure is a length of pipe that we have produced by cutting very long pipes in a factory, and let us further suppose that if a pipe is too short (i.e., less than our hypothetical confidence interval lower bound) we must discard it and that costs twice as much as a pipe that is too long, which too long pipe can be sent back for trimming to length. In that case, we might want a left tail that is half as probable as a right tail in order to balance the cost of having pipes of the wrong length.
