Question

2. Suppose over several years of offering AP Statistics, a high school finds that final exam scores are normally distributed with a mean of 78 and a standard deviation of 6. A. What are the mean, standard deviation, and shape of the distribution of x-bar for n

Answer 1

By the Central Limit Theorem, the mean is 78, the standard deviation is
`s = (6)/(√(n))` and the shape is approximately normal.

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.

Mean of 78 and a standard deviation of 6

This means that
`\mu = 78, \sigma = 6`

Samples of n:

This means that the standard deviation is:

`s = (\sigma)/(√(n)) = (6)/(√(n))`

What are the mean, standard deviation, and shape of the distribution of x-bar for n?

By the Central Limit Theorem, the mean is 78, the standard deviation is
`s = (6)/(√(n))` and the shape is approximately normal.