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.
(a) Test, at the \(10 \%\) significance level, whether Jade's claim is justified.

(b) State an assumption that is necessary for this test to be valid.