3 An engineering company buys a certain type of component from two suppliers, A and B. It is important that, on the whole, the strengths of these components are the same from both suppliers. The company can measure the strengths in its laboratory. Random samples of seven components from supplier A and five from supplier B give the following strengths, in a convenient unit. The underlying distributions of strengths are assumed to be Normal for both suppliers, with variances 2.45 for supplier A and 1.40 for supplier B.

1. Test at the \(5 \%\) level of significance whether it is reasonable to assume that the mean strengths from the two suppliers are equal.
2. Provide a two-sided 90\% confidence interval for the true mean difference.
3. Show that the test procedure used in part (i), with samples of sizes 7 and 5 and a \(5 \%\) significance level, leads to acceptance of the null hypothesis of equal means if \(- 1.556 < \bar { x } - \bar { y } < 1.556\), where \(\bar { x }\) and \(\bar { y }\) are the observed sample means from suppliers A and B . Hence find the probability of a Type II error for this test procedure if in fact the true mean strength from supplier A is 2.0 units more than that from supplier B.
4. A manager suggests that the Wilcoxon rank sum test should be used instead, comparing the median strengths for the samples of sizes 7 and 5 . Give one reason why this suggestion might be sensible and two why it might not.