10 Carpal Tunnel syndrome is a condition which affects a person's ability to grip with their hands. Researchers tested a treatment for this syndrome which was applied to 8 randomly chosen patients. A pre-treatment and a post-treatment test of grip was given to each patient, with the following results, measured in kg.
| Patient | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Pre-treatment grip | 24.3 | 29.5 | 28.0 | 28.5 | 21.5 | 28.7 | 25.1 | 26.3 |
| Post-treatment grip | 28.3 | 34.6 | 30.3 | 31.6 | 21.5 | 29.8 | 26.0 | 27.5 |
Stating any required assumption, test, at the \(1 \%\) significance level, whether the mean grip of people with the syndrome increases after undergoing the treatment.
It is given that there is evidence at the \(10 \%\) significance level that the mean grip increases by more than \(w \mathrm {~kg}\). Find an inequality for \(w\).