Question

A simple random sample of size nequals10 is obtained from a population with muequals63 and sigmaequals18. ​(a) What must be true regarding the distribution of the population in order to use the normal model to compute probabilities involving the sample​ mean? Assuming that this condition is​ true, describe the sampling distribution of x overbar.

Answer 1

The sample size is smaller than 30, so we need to assume that the underlying population is normally distributed.

The sampling distribution of x overbar will be approximately normally distributed with mean 63 and standard deviation 5.69.

Explanation:

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

In this problem

The sample size is smaller than 30, so we need to assume that the underlying population is normally distributed.

If it is:

`\mu = 63, \sigma = 18, n = 10, s = (18)/(√(10)) = 5.69`

The sampling distribution of x overbar will be approximately normally distributed with mean 63 and standard deviation 5.69.