Question

The height of American adult women is distributed almost exactly as a normal distribution. The mean height of adult American women is 63.5 inches with a standard deviation of 2.5 inches. Imagine that all possible random samples of size 25(n = 25) are taken from the population of American adult women's heights. The means from each sample would then be graphed to form the sampling distribution of sample means. The mean of this sampling distribution is _____ and the standard deviation of this sampling distribution is ________.

Answer 1

The mean of this sampling distribution is 63.5 and the standard deviation of this sampling distribution is 0.5

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, we have that:

`\mu = 63.5, \sigma = 2.5, n = 25, s = (2.5)/(√(25)) = 0.5`

So the correct answer is:

The mean of this sampling distribution is 63.5 and the standard deviation of this sampling distribution is 0.5