Problem

Jasmine is researching the heights of pine trees in forests in two regions $A$ and $B$. She chooses a random sample of 50 pine trees in region $A$ and records their heights, $x$ m. She also chooses a random sample of 60 pine trees in region $B$ and records their heights, $y$ m. Her results are summarised as follows.

$$\begin{gathered} \sum x = 1625,\qquad \sum x^2 = 53200,\\ \sum y = 1854,\qquad \sum y^2 = 57900. \end{gathered}$$

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