1 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. $$\sum x = 6345 \quad \sum ( x - \bar { x } ) ^ { 2 } = 243.8 \quad \sum y = 7614 \quad \sum ( y - \bar { y } ) ^ { 2 } = 384.9$$ \(\bar { x }\) 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 \(Y\).