Question

A random sample of n=70 measurements is obtained from a population with a mean 200 and standard deviation of 50.describe the sampling for the sample means by computing

Answer 1

The Central Limit Theorem

For samples of size 30 or more, the sample mean is approximately normally distributed, with mean $\mu_{\bar{X}}=\mu$ and standard deviation $\sigma_{\bar{X}}=\sigma/\sqrt{n},$ where $n$  is the sample size.

The larger the sample size, the better the approximation.

$$\bar{X}\sim N(\mu, \sigma^2/n)$$

Given $\mu=200, \sigma=50, n=70.$

The distribution of the sample mean is normal with mean $\mu_{\bar{X}}=\mu=200$ and standard deviation $\sigma_{\bar{X}}=\sigma/\sqrt{n}=50/\sqrt{70}$

$$\bar{X}\sim N(200, 50^2/70)$$