Question

A normally distributed population has a mean of 40 and a standard deviation of 12. What does the Central Limit Theorem say about the sampling distribution of the mean if samples of size 100 are drawn from this population?

Answer 1

see below

Explanation:

If X is the random variable for the Normal distribution then

`X~N(40,12^2)`

for sample sizes of 100 the sampling distribution of the mean follows the following distribution, according to the Central Limit Theorem

`barX~N(40,12^2/100)`

the Central Limit theorem is useful because it doesn't matter what the, background distribution is providing the sample size, `n>=30` the distribution of the sample mean will follow approximately a Normal distribution

`barX~N(mu,sigma^2/n)`