Question

A simple random sample of size nequals57 is obtained from a population with muequals69 and sigmaequals2. Does the population need to be normally distributed for the sampling distribution of x overbar to be approximately normally​ distributed? Why? What is the sampling distribution of x overbar​?

Answer 1

Answer:

The population does not need to be normally distributed for the sampling distribution of $\bar{X}$ to be approximately normally distributed. Because of the central limit theorem. The sampling distribution of $\bar{X}$ is approximately normal.

Step-by-step explanation:

We have a random sample of size $n = 57$ from a population with $\mu = 69$ and $\sigma = 2$. Because n is large enough (i.e., n > 30) and $\mu$ and $\sigma$ are both finite, we can apply the central limit theorem that tell us that the sampling distribution of $\bar{X}$ is approximatelly normally distributed, this independently of the distribution of the random sample. $\bar{X}$ is asymptotically normally distributed is another way to state this.