6. An engineer has developed a new battery. She claims that the new battery will last more than 8 hours longer, on average, than the old battery. To test the claim, the engineer randomly selects a sample of 50 new batteries and 40 old batteries. She records how long each battery lasts, \(x\) hours for the new batteries and \(y\) hours for the old batteries. The results are summarised in the table below.
| \cline { 2 - 4 }
\multicolumn{1}{c|}{} | \(n\) | Sample mean | \(s ^ { 2 }\) |
| New battery | 50 | \(\bar { x } = 83\) | 7 |
| Old battery | 40 | \(\bar { y } = 74\) | 6 |
- Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the engineer's claim. State your hypotheses and show your working clearly.
- Explain the relevance of the Central Limit Theorem to the test in part (a).