Dev owns a small company which produces bottles of juice. He uses two machines, $$\(X\)$$ and $$\(Y\)$$, to fill empty bottles with juice. Dev is investigating the volumes of juice in the bottles. He chooses a random sample of 35 bottles filled by machine $$\(X\)$$ and a random sample of 60 bottles filled by machine $$\(Y\)$$. The volumes of juice, $$\(x\)$$ and $$\(y\)$$ respectively, measured in suitable units, are summarised by $$\[ \Sigma x=30.8, \quad \Sigma x^{2}=29.0, \quad \Sigma y=62.4, \quad \Sigma y^{2}=76.8 . \]$$ Dev claims that the mean volume of juice in bottles filled by machine $$\(Y\)$$ is greater than the mean volume of juice in bottles filled by machine $$\(X\)$$. A test at the $$\(\alpha \%\)$$ significance level suggests that there is sufficient evidence to support Dev's claim. 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