4. A coach believes that the average score in the final round of a golf tournament is more than one point below the average score in the first round. To test this belief, the scores of 8 randomly selected players are recorded. The results are given in the table below.
| Player | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) |
| First round | 76 | 80 | 72 | 78 | 83 | 88 | 81 | 72 |
| Final round | 70 | 78 | 75 | 75 | 79 | 84 | 83 | 69 |
- State why a paired \(t\)-test is suitable for use with these data.
- State an assumption that needs to be made in order to carry out a paired \(t\)-test in this case.
- Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the coach's belief. Show your working clearly.
- Explain, in the context of the coach's belief, what a Type II error would be in this case.