Problem

A factory produces small bottles of natural spring water. Two different machines, $X$ and $Y$, are used to fill empty bottles with the water. A quality control engineer checks the volumes of water in the bottles filled by each of the machines. He chooses a random sample of 60 bottles filled by machine $X$ and a random sample of 75 bottles filled by machine $Y$. The volumes of water, $x$ and $y$ respectively, in millilitres, are summarised as follows.

$$\begin{gathered} \sum x = 6345,\qquad \sum (x-\bar{x})^2 = 243.8,\\ \sum y = 7614,\qquad \sum (y-\bar{y})^2 = 384.9. \end{gathered}$$

$\bar{x}$ and $\bar{y}$ are the sample means of the volume of water in the bottles filled by machines $X$ and $Y$ respectively.

Find a 95% confidence interval for the difference between the mean volume of water in bottles filled by machine $X$ and the mean volume of water in bottles filled by machine $Y$.