Question

A random sample of n=90| kilograms of corn is obtained from a population with u =272 and sigma = 5543 . Describe the sampling distribution for the sample means by computing the mu hat x and sigma_{x} (Use the indefinite population)

Answer 1

The Central Limit Theorem states that if you have a population with mean $\mu$ and standard deviation $\sigma$ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large (usually $n > 30$ ).

We have $n=90>30.$

Then by the Central Limit Theorem $\bar{X}\sim N(\mu, \sigma^2/n)$

$$\mu_{\bar{X}}=\mu=272$$

$$\sigma_{\bar{X}}=\dfrac{\sigma}{\sqrt{n}}=\dfrac{5543}{\sqrt{90}}\approx584.2835$$