Question

A random variable x has a Normal distribution with an unknown mean μ and a standard deviation σ=12. Suppose that we take a random sample of size n=36 and find a sample mean of x¯=98.

Answer 1

The sample mean of
`\bar x` = 98 is 1.

The Central Limit Theorem (CLT) states that for a random sample of size n from a distribution with mean μ and standard deviation σ, the sample mean
`\bar x` will have an approximate normal distribution with mean μ and standard deviation σ/√n.

In this case, we have n=36 and σ=12, so the standard deviation of the sample mean is σ/√n = 12/√36 = 2.

Therefore, the sample mean
`\bar x` = 98 is approximately normally distributed with mean μ=98 and standard deviation 2.

This means that we can use the normal distribution to calculate probabilities related to the sample mean.

For example, we can calculate the probability that the sample mean is greater than 100:

P(
`\bar x` > 100) = P(z > 1) = 0.1587

where z is the standard normal variable defined by

z = (
`\bar x` - μ) / (σ/√n) = (100 - 98) / (2) = 1