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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?

eugene
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  • The true value of $\mu$ is not used when constructing a confidence interval for the mean, so you don't need to know that. If the sample size $n$ is large, then the CLT implies that $Y=\sqrt{n}(\bar X - \mu)/S$ has approximately a standard normal distribution, so you can obtain an approximate confidence interval. (But I can't say how good the approximation is, so that is somewhat hand-wavey.) If $n$ is small but your sample is from a normal distribution, then $Y$ has a $t$-distribution with $n-1$ degrees of freedom. So in that case the theory is nice and you can get a confidence interval. – littleO Feb 16 '19 at 06:10

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