Question

A random sample of size 36 is to be taken from a population that is normally distributed with mean 72 and standard deviation 6. The sample mean of the observations in our sample is to be computed. The sampling distribution of the sample mean is

Answer 1

The sampling distribution of the sample mean is approximately normal with mean 72 and standard deviation 1.

Explanation:

Central Limit Theorem

The Central Limit Theorem establishes 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.

Normally distributed with mean 72 and standard deviation 6.

This means that
`\mu = 72, \sigma = 6`

A random sample of size 36

This means that
`n = 36, s = (6)/(√(36)) = 1`

The sampling distribution of the sample mean is

By the Central Limit Theorem, it is approximately normal with mean 72 and standard deviation 1.