Mean and standard deviation of difference of sample means

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 mm

with a standard deviation of

0.003 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:

σ_{{\bar{x}}_{1} - {\bar{x}}_{2}} = \sqrt{\frac{σ_{1}^{2}}{n_{1}} + \frac{σ_{2}^{2}}{n_{2}}}

Why is it not appropriate to use this formula for the standard deviation of ${\bar{x}}_{A} - {\bar{x}}_{B}$ ‍?

AP®︎/College Statistics