4 An inspector is checking the lengths of metal rods produced by two machines, \(X\) and \(Y\). These rods should be of the same length, but the inspector suspects that those made by machine \(X\) are shorter, on average, than those made by machine \(Y\). The inspector chooses a random sample of 80 rods made by machine \(X\) and a random sample of 60 rods made by machine \(Y\). The lengths of these rods are \(x \mathrm {~cm}\) and \(y \mathrm {~cm}\) respectively. Her results are summarised as follows. $$\sum x = 164.0 \quad \sum x ^ { 2 } = 338.1 \quad \sum y = 124.8 \quad \sum y ^ { 2 } = 261.1$$

1. Test at the \(10 \%\) significance level whether the data supports the inspector's suspicion.
2. Give a reason why it is not necessary to make any assumption about the distributions of the lengths of the rods.