A scientist is investigating the lengths of the leaves of birch trees in different regions. He takes a random sample of 50 leaves from birch trees in region \(A\) and a random sample of 60 leaves from birch trees in region \(B\). He records their lengths in \(\mathrm{cm}, x\) and \(y\), respectively. His results are summarised as follows.
\(\sum x=282 \quad \sum x^{2}=1596 \quad \sum y=328 \quad \sum y^{2}=1808\)

The population mean lengths of leaves from birch trees in regions \(A\) and \(B\) are \(\mu_{A} \mathrm{~cm}\) and \(\mu_{B} \mathrm{~cm}\) respectively.

Carry out a test at the \(5 \%\) significance level to test the null hypothesis \(\mu_{A}=\mu_{B}\) against the alternative hypothesis \(\mu_{A} \neq \mu_{B}\).