7 In order to improve their mathematics results 10 students attended an intensive Summer School course. Each student took a test at the start of the course and a similar test at the end of the course. The table shows the scores achieved in each test.
| Student | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| First test score | 37 | 27 | 38 | 47 | 54 | 27 | 52 | 39 | 62 | 23 |
| Second test score | 47 | 29 | 50 | 44 | 72 | 37 | 63 | 45 | 76 | 32 |
It is desired to test whether there has been an increase in the population mean score.
- Explain why a two-sample \(t\)-test would not be appropriate.
- Stating any necessary assumptions, carry out a suitable \(t\)-test at the \(\frac { 1 } { 2 } \%\) significance level.
- The Summer School director claims that after taking the course the population mean score increases by more than 5 . Is there sufficient evidence for this claim?