Problem 4.

Suppose a sample of size n = 100 is taken from a population with mean $\mu$ = 70 and standard deviation $\sigma$ = 20. Let $\bar{X}$ denote the sample mean.

(a) What is the mean and standard deviation of $\bar{X}$?

Solution: We have

$$E[\bar{X}] \; = \; \mu \; = \; 70 ,$$

and

$$V[\bar{X}] \; = \; \sigma^2/n ,$$

so

$$\sigma_{\bar{X}} \; = \; \sigma/\sqrt{n} \; = \; 20/\sqrt{100} \; = \; 20/10/ \; = \; 10 .$$

(b) What is the approximate distribution of $\bar{X}$?

Solution: By the Central Limit Theorem, $\bar{X}$ is approximately normally distributed with mean 70 and standard deviation 2.