3 A tutor gave an assessment to 6 randomly chosen new eleven-year-old students. After each student had received 30 hours of tuition, they were given a second assessment. The scores are shown in the table.
| Student | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) |
| 1st assessment | 124 | 121 | 111 | 113 | 118 | 119 |
| 2nd assessment | 127 | 119 | 114 | 110 | 120 | 122 |
- Show that, at the \(5 \%\) significance level, there is insufficient evidence that students' scores are higher, on average, after tuition than before tuition. State a necessary assumption.
- Disappointed by this result, the tutor looked again at the first assessment. She discovered that the first assessment was too easy, in fact being a test for ten-year-olds, not eleven-year-olds. She decided to reduce each score for the first assessment by a constant integer \(k\). Find the least value of \(k\) for which there is evidence at the \(5 \%\) significance level that the students' scores have, on average, improved.