The change of the distribution of a sample mean when n reaches infinity

Question

Hi guys, I posted this question yesterday, but I deleted it because I didn't clarify it. Now I am posting with the picture attached. Is the following correct?

I think it is correct, because if the sample is really large, then the distribution of the mean would converge to the expected value. But how to elaborate it in a more formal way? Or maybe someone can prove that it is not true. Thank you in advance.

Answer 1

No, it's not true.

The normal distribution can take on an infinite number of values, yet the sample average for a rando sample drawn from a population whose measurements follow this distribution (e.g., height to a good approximation) will have non-zero variance.

The question is not about the size of the sample, but about the size of the sample space. Two completely different concepts. Your statement about large sample properties is generally correct for distributions with finite mean. However, there is no "Law of Large Sample Spaces" that would make the statement true.

Answer 2

It is true.

For the sample you'll get some standard deviation, $\sigma$.

The standard deviation of the mean is $${\sigma}_m = \dfrac{\sigma}{n^{0.5}}$$

where $n$ is the sample size. So as $n$ goes to infinity the standard deviation of the mean converges to zero.