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

Answer:

By the Central Limit Theorem, the mean is 78, the standard deviation is $s = \frac{6}{\sqrt{n}}$ and the shape is approximately normal.

Step-by-step 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 $s = \frac{\sigma}{\sqrt{n}}$.

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:

$s = \frac{\sigma}{\sqrt{n}} = \frac{6}{\sqrt{n}}$

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 $s = \frac{6}{\sqrt{n}}$ and the shape is approximately normal.