1 Ellie is investigating the heights of two types of beech tree, \(A\) and \(B\), in a certain region. She has chosen a random sample of 60 beech trees of type \(A\) in the region, recorded their heights, \(x \mathrm {~m}\), and calculated unbiased estimates for the population mean and population variance as 35.6 m and \(4.95 \mathrm {~m} ^ { 2 }\) respectively. Ellie also chooses a random sample of 50 beech trees of type \(B\) in the region and records their heights, \(y \mathrm {~m}\). Her results are summarised as follows. $$\sum y = 1654 \quad \sum y ^ { 2 } = 54850$$ Find a \(95 \%\) confidence interval for the difference between the population mean heights of type \(A\) and type \(B\) beech trees in the region.