8 Two machines, \(A\) and \(B\), produce metal rods. Machine \(B\) is new and it is required that its accuracy should be checked against that of machine \(A\). The observed variable is the length of a rod. Random samples of rods, 40 from machine \(A\) and 50 from machine \(B\), are taken and their lengths, \(x _ { A } \mathrm {~cm}\) and \(x _ { B } \mathrm {~cm}\), are measured. The results are summarised by $$\Sigma x _ { A } = 136.48 , \quad \Sigma x _ { B } = 176.35 , \quad \Sigma x _ { B } ^ { 2 } = 630.1940 .$$ The variance of the length of the rods produced by machine \(A\) is known to be \(0.0490 \mathrm {~cm} ^ { 2 }\). The mean lengths of the rods produced by the machines are denoted by \(\mu _ { A } \mathrm {~cm}\) and \(\mu _ { B } \mathrm {~cm}\) respectively.

1. Test, at the \(5 \%\) significance level, the hypothesis \(\mu _ { B } > \mu _ { A }\).
2. Find the set of values of \(a\) for which the null hypothesis \(\mu _ { B } - \mu _ { A } = 0.025\) would not be rejected in favour of the alternative hypothesis \(\mu _ { B } - \mu _ { A } > 0.025\) at the \(a \%\) significance level.
3. For the test in part (i) to be valid,
   (a) state whether it is necessary to assume that the two population variances are equal,
   (b) state, giving a reason, whether it is necessary to assume that the lengths of rods are normally distributed.