Confidence intervals and assumption of normality

Hello,

I am analysing some data and calculating the 95% confidence intervals with the 'usual' approach of Xbar +/- t * s/sqrt(n).

However I have read several times that one of the assumptions using this method is that the distribution has to be normal (or close to normal). My *original* data is far from normal and is noticably skewed right. HOWEVER my understanding is that providing my sample size is >30 (in most cases it is) then the *sampling* distribution will be normal. Since it is the sampling distribution I am using to calculate the confidence interval then the original sample distribution does not have to be normal.

Does anyone know if this is correct? I.e. that as long as n>30 it does not matter what shape the original distribution is since my sampling distribution will be normal (or close to normal) so I can use the above method to calculate the 95% CI's?

Cheers!

Rob