4 William takes a bus regularly on the same journey, sometimes in the morning and sometimes in the afternoon. He wishes to compare morning and afternoon journey times. He records the journey times on 7 randomly chosen mornings and 8 randomly chosen afternoons. The results, each correct to the nearest minute, are as follows, where M denotes a morning time and A denotes an afternoon time.
| M | A | A | M | M | M | M | M | M | A | A | A | A | A | A |
| 19 | 20 | 22 | 24 | 25 | 26 | 28 | 30 | 31 | 33 | 35 | 37 | 38 | 39 | 42 |
William wishes to test for a difference between the average times of morning and afternoon journeys.
- Given that \(s _ { M } ^ { 2 } = 16.5\) and \(s _ { A } ^ { 2 } = 64.5\), with the usual notation, explain why a \(t\)-test is not appropriate in this case.
- William chooses a non-parametric test at the \(5 \%\) significance level. Carry out the test, stating the rejection region.