Problem

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, $x$ m, and calculated unbiased estimates for the population mean and population variance as $35.6$ m and $4.95$ m$^2$ respectively.

Ellie also chooses a random sample of 50 beech trees of type $B$ in the region and records their heights, $y$ m. Her results are summarised as follows.

$$\sum y = 1654 \qquad \sum y^2 = 54\,850$$

Find a 95% confidence interval for the difference between the population mean heights of type $A$ and type $B$ beech trees in the region.