# Sampling distribution of a sample mean example

Here's the type of problem you might see on the AP Statistics exam where you have to use the sampling distribution of a sample mean.

## Example: Means in quality control

An auto-maker does quality control tests on the paint thickness at different points on its car parts since there is some variability in the painting process. A certain part has a target thickness of $2\text{ mm}$. The distribution of thicknesses on this part is skewed to the right with a mean of $2\text{ mm}$ and a standard deviation of $0.5\text{ mm}$.
A quality control check on this part involves taking a random sample of $100$ points and calculating the mean thickness of those points.
Assuming the stated mean and standard deviation of the thicknesses are correct, what is the probability that the mean thickness in the sample of $100$ points is within $0.1\text{ mm}$ of the target value?
Let's solve this problem by breaking it down into smaller parts.

### Part 1: Establish normality

What is the shape of the sampling distribution of the sample mean thickness?

### Part 2: Find the mean and standard deviation of the sampling distribution

The sampling distribution of a sample mean $\bar x$ has:
\begin{aligned} \mu_{\bar x}&=\mu \\\\ \sigma_{\bar x}&=\dfrac{\sigma}{\sqrt n} \end{aligned}
Note: For this standard deviation formula to be accurate, our sample size needs to be $10\%$ or less of the population so we can assume independence.
Question A (Part 2)
What is the mean of the sampling distribution of $\bar x$?
$\mu_{\bar x}=$
$\text{mm}$

Question B (Part 2)
What is the standard deviation of the sampling distribution of $\bar x$?
$\sigma_{\bar x}=$
$\text{mm}$

### Part 3: Use normal calculations to find the probability in question

Assuming the stated mean and standard deviation of the thicknesses are correct, what is the approximate probability that the mean thickness in the sample of $100$ points is within $0.1\text{ mm}$ of the target value?