5. A doctor claims there is a higher mean lung capacity in people who exercise regularly compared to people who do not exercise regularly. He measures the lung capacity, \(x\), of 35 people who exercise regularly and 42 people who do not exercise regularly. His results are summarised in the table below.
| \cline { 2 - 4 }
\multicolumn{1}{c|}{} | \(n\) | \(\bar { x }\) | \(s ^ { 2 }\) |
| Exercise regularly | 35 | 26.3 | 12.2 |
| Do not exercise regularly | 42 | 24.8 | 10.1 |
- Test, at the \(5 \%\) level of significance, the doctor's claim. State your hypotheses clearly.
- State any assumptions you have made in testing the doctor's claim.
The doctor decides to add another person who exercises regularly to his data. He measures the person's lung capacity and finds \(x = 31.7\)
- Find the unbiased estimate of the variance for the sample of 36 people who exercise regularly. Give your answer to 3 significant figures.