# 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

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

The sampling distribution of a sample mean $\bar x$ has:

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.