9231 P42 - Nov 2023 - Q5 - 16 marks
A company is deciding which of two machines, \(X\) and \(Y\), can make a certain type of electrical component more quickly. The times taken, in minutes, to make one component of this type are recorded for a random sample of 8 components made by machine \(X\) and a random sample of 9 components made by machine \(Y\). These times are as follows.
Machine \(X\) | 4.0 | 4.6 | 4.7 | 4.8 | 5.0 | 5.2 | 5.6 | 5.8 | |
|---|---|---|---|---|---|---|---|---|---|
Machine \(Y\) | 4.5 | 4.9 | 5.1 | 5.3 | 5.4 | 5.7 | 5.9 | 6.3 | 6.4 |
The manager claims that on average the time taken by machine \(X\) to make one component is less than that taken by machine \(Y\).
(a) Carry out a Wilcoxon rank-sum test at the \(5 \%\) significance level to test whether the manager's claim is supported by the data.
(b) Assuming that the times taken to produce the components by the two machines are normally distributed with equal variances, carry out a \(t\)-test at the \(5 \%\) significance level to test whether the manager's claim is supported by the data.
(c) In general, would you expect the conclusions from the tests in parts (a) and (b) to be the same? Give a reason for your answer.