Question

if the variance of the population is 9, what happens to the variance of the sampling distribution of the means of size 6 drawn from the population?

Answer 1

Central Limit Theorem: If $\bar{X}$ is the mean of a random sample of size $n$ taken

from a population with mean $\mu$ and finite variance $\sigma^2,$ then the limiting form of the distribution of

$$Z=\dfrac{\bar{X}-\mu}{\sigma/\sqrt{n}}$$

as $n\to \infin,$ , is the standard normal distribution $n(z; 0, 1)$

Given $\sigma^2=9, n=6.$

Then the variance of the sampling distribution of the means of size 6 drawn from the population will be

$$\sigma^2/n=9/6=1.5$$