Answer to Question #197136 in Statistics and Probability for sharmay genita

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

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