This is an interesting question. If you're using the sample mean (of the group) as the point estimator for the true mean, it has an approximately **normal** sampling distribution. This is often the case, so if your 95% CI is 1 <= mu <= 2, then you should have Xbar = 1.5 (right in the middle of the CI).

You could find other CIs, for example 90% CI may be 1.2 <= mu <= 1.8. Then there is a 5% chance that mu is in [1, 1.2] or [1.8, 2]--this is a total interval length of 0.2 + 0.2 = 0.4 (compare to 90% chance it's in the interval of length 0.6 from [1.2, 1.8]). These are just made-up numbers, but that's my idea.