I understand https://math.stackexchange.com/a/192910/291153 and I could rephrase it something like :
The true mean is likely to fall under the confidence interval x% times if we do the experiment many times.
But isn't the statement true only when we use the true mu and true sigma? We are using the estimates from the sample to get the value of intervals.
Isn't it more accurate to describe confidence interval as :
It's a guess (it's a best guess we could do) that The true mean is likely to fall under the confidence interval x% times if we do the experiment many times
I don't know how to precisely (academically) state the It's a guess part, but just wonder if it is a legitimate suspicion?