The Sampling Distribution of the Sample Mean

Theorem

Let be IID with , then

(see summation notation)

Expected value:

Variance:

Standard deviation (aka standard error):

Note: we have an alternate z-score formulation for sample mean (as opposed to population mean):

Intuitively, this makes sense. When , we just have the old formula, this is just a bit of a generalization.

Example

Suppose that final grades in this course follow a normal distribution with and . A sample of 9 students is taken. Find:

solution

P(S \ge 700) & = P\left( \frac{s - 675}{\sqrt{ 576 }} \ge \frac{700 - 675}{\sqrt{ 576 }} \right) \
& = P(Z \ge 1.04) \
& = 1 - 0.85083
\end