If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## AP®︎/College Statistics

### Course: AP®︎/College Statistics>Unit 9

Lesson 7: Sampling distributions for differences in sample means

# Mean and standard deviation of difference of sample means

## Problem

A car manufacturer has two production plants in different cities. Every day, plant A produces $120$ of a certain type of car, while plant B produces $80$ of those cars. On average, all of these cars have a paint thickness of $0.04\phantom{\rule{0.167em}{0ex}}\text{mm}$ with a standard deviation of $0.003\phantom{\rule{0.167em}{0ex}}\text{mm}$.
Every day, quality control experts take separate random samples of $10$ cars from each plant and calculate the mean paint thickness for each sample. They then look at the difference between those sample means.
Consider the formula:
${\sigma }_{{\overline{x}}_{1}-{\overline{x}}_{2}}=\sqrt{\frac{{\sigma }_{1}^{2}}{{n}_{1}}+\frac{{\sigma }_{\text{2}}^{2}}{{n}_{2}}}$
Why is it not appropriate to use this formula for the standard deviation of ${\overline{x}}_{\text{A}}-{\overline{x}}_{\text{B}}$?