People get this confused all the time.

You use Z, when you have

If the underlying population is normal, i.e.,

, then

BUT in real life, how would we know the st deviation when we usually don't know the .

So if we estimate with and ....

the underlying population is normal, i.e.,

, then

---------------------------------------------------------------------

NOW for the part of having a large sample.

This n larger than 30 I assume is from the Berry-Esseen Theorem, which

tells you how close you are to a normal distribution.

http://en.wikipedia.org/wiki/Berry%E...Esseen_theorem

If you have i.i.d random variables, then both

and

approach a standard normal random variable as n goes to INFINITY (CLT).

Now, we never sample an infinite number of items, but the larger the better.

Note that a t density approaches a normal as n goes to infinity,

so it doesn't make much difference which table you use for very large n.