# How can we estimate sampling distribution's mean and std using only one sample's mean and std

Hi everyone
I’m having some trouble with the exercise shown in the image below, and I can’t stop thinking about it.

MY UNDERSTANDING
This exercise revolves around the Central Limit Theorem. As I understand, this theorem works as follows: You want to determine the mean and standard deviation of a population distribution, but this is too difficult to determine directly. So, you need to do it indirectly. You perform this indirect method by collecting multiple samples, each with a sufficiently large sample size (the observation in a sample - usually 30). Each of these samples will give you a sample mean. By gathering all the sample means, you create SAMPLING DISTRIBUTION (a distribution of the sample means). This sampling distribution will have a mean = population’s mean and a std = population’s std/√n (n is the sample size – the number of observations in a sample). This isn’t exactly what the Central Limit Theorem states, but it is how I understand it works.

THE PROBLEM
So, to determine the sampling distribution, you need a sufficiently large number of samples, each with a sufficiently large sample size. However, in many exercises I’ve encountered, things don’t seem to work this way. Instead of collecting multiple samples, they only determine ONE SINGLE SAMPLE with its mean and standard deviation. THEY USE THAT SINGLE SAMPLE’S MEAN AND STD TO ESTIMATE THE SAMPLING DISTRIBUTION’S MEAN AND STD. This really confuses me; how can they estimate an entire sampling distribution based on just one sample?

Additionally, the explanation at the end of the exercise also confuses me. Why does it have to be a “all the combination of samples of size 100”? Could it be a combination of samples of size 70/80/90/110/120, etc., or does it have to be exactly 100?

SO MY QUESTION IS:
(1) Why can they use one single sample to estimate the sampling distribution?
(2) Why is the final explanation in the exercise stated that way? Could I apply it to other sample sizes besides 100 or it has to be 100?