Math Help - Student T-Distribution or Gaussian Distribution

1. Student T-Distribution or Gaussian Distribution

Hey guys

Ok the background is this, I have 1500 observation and after I "compressed" the data using a stat package, I was given 20 observation.

The question is, should i be using the Gaussian distribution or the t-distribution with 20dfs? And why?

Thanks!

2. IT does NOT depend on the sample size
IF you know $\sigma$ then you use the normal curve
IF you do NOT know $\sigma$ then you replace sigma with s and use a t
IN both cases I am assuming that the underlying distribution is normal.
NOW if the sample size is large, you can use the central limit theorem to make the test/CI approximately normal.
And it's n-1=19 dfs

3. Thanks a lot!!!

But the standard deviation/variance. Can't we just use the formula and estimate it?

4. Of course, you use s to estimate sigma.
Then the dirstibution of

${\bar X-\mu\over s/\sqrt{n}}$ is a t with n-1 dfs

and if you know sigma then

${\bar X-\mu\over \sigma/\sqrt{n}}$ is a N(0,1)

where I'm assuming that the data is from a Normal.
NOW If the sample size is large, either will be close to N(0,1) no matter if you use sigma or s
since the CLT is taking over and s approaches sigma with probability one.