Consider a random sample from a Weibull distribution, $X_i \sim WEI(1,2)$. Find approximate values $a$ and $b$ such that for $n=35$:
(a) $P[a < \bar{X} < b]=0.95$
(b) $P[a < X_{18:35} < b] = 0.95$
Here $\bar{X}=\frac{X_1+\cdots+X_n}n$ (the sample mean) and $X_{18:35}$ is the sample median. Also the pdf of $X_i$ is $2xe^{-x^2}$. We need $P[\bar{X}<b]-P[\bar{X}<a]=0.95$. But is $(a,b)$ unique? If so, why? I think it's not unique.
You are right, $a$ and $b$ are not uniquely determined. But if we decide to make $a$ and $b$ as symmetric as possible about the mean, they are determined.
In this case, the mean is $\frac{\sqrt{\pi}}{2}$. The numbers you mentioned in a comment are in fact chosen to be symmetric about this.