6 Jade is a swimming instructor at a sports college. She claims that, as a result of an intensive training course, the mean time taken by students to swim 50 metres has reduced by more than 1 second. She chooses a random sample of 10 students. The times taken, in seconds, before and after the training course are recorded in the table.
| Student | A | B | C | D | \(E\) | \(F\) | G | \(H\) | I | J |
| Time before course | 54.2 | 47.4 | 52.1 | 59.0 | 55.3 | 51.0 | 48.9 | 52.2 | 58.4 | 51.4 |
| Time after course | 50.1 | 46.3 | 52.5 | 58.8 | 51.4 | 48.4 | 49.5 | 48.7 | 58.3 | 51.4 |
- Test, at the 10\% significance level, whether Jade's claim is justified.
- State an assumption that is necessary for this test to be valid.