4 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 \(\mu _ { \mathrm { A } } = \mu _ { \mathrm { B } }\) against the alternative hypothesis \(\mu _ { \mathrm { A } } \neq \mu _ { \mathrm { B } }\).