Why One Sample For Confidence Intervals?

#1
As I understand it, The Central Limit Theorem and Confidence Intervals are related. What is the best way to explain why the Confidence Interval formula uses only one sample whereas the Central Limit Theorem uses 'many' samples? Isn't the x-bar in the CI formula the mean of one sample? Don't we take many samples to explain the CLT?
Is the best thing that we can say is that a 95% CI means that if we took say 100 samples then 95 of them would contain the mean.
 

noetsi

Fortran must die
#2
I don't think the Central Limit Theorem uses any samples. Its a theory that says something will occur were you to use many samples. The confidence interval is being estimated from a sample for an unknown true population. It can't be estimated from multiple samples (unless you combined them into one or they had the same sample standard deviation which is very unlikely with real data).

'Is the best thing that we can say is that a 95% CI means that if we took say 100 samples then 95 of them would contain the mean." This might be true using a Bayesian approach. Its not literally what frequentist, the most common type of statistics you learn in school, mean by this as far as I know. The two schools of statistics answer this question differently.
 
#3
One of many descriptions of the CLT follow: "The Central Limit Theorem (CLT for short) basically says that for (even) non-normal data, the distribution of the sample means has an approximate normal distribution, no matter what the distribution of the original data looks like, as long as the sample size is large enough (usually at least 30) and all samples have the same size" How can I relate this definition to Confidence Intervals? P.S. I added the word 'even.'
 

noetsi

Fortran must die
#4
To me the CLT (a theory) is very different from the CI (a calculation). CI require assumptions about the distribution, commonly a normal distribution is assumed. Because of the CLT, if the sample size is large enough even if the population is non-normal you can still calculate a CI which relies on normality correctly I think.