Question

The lengths of adult males' hands are normally distributed with mean 187 mm and standard deviation is 7.1 mm. Suppose that 12 individuals are randomly chosen. Round all answers to 4 where possible. What is the distribution of ¯ x

Answer 1

By the Central Limit Theorem, the distribution of ¯ x is approximately normal with mean 187 and standard deviation 2.05.

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.

The lengths of adult males' hands are normally distributed with mean 187 mm and standard deviation is 7.1 mm.

This means that
`\mu = 187, \sigma = 7.1`

Suppose that 12 individuals are randomly chosen.

This means that
`n = 12, s = (7.1)/(√(12)) = 2.05`

What is the distribution of ¯ x?

By the Central Limit Theorem, the distribution of ¯ x is approximately normal with mean 187 and standard deviation 2.05.