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

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.

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.