6. A sociologist was studying the smoking habits of adults. A random sample of 300 adult smokers from a low income group and an independent random sample of 400 adult smokers from a high income group were asked what their weekly expenditure on tobacco was. The results are summarised below.
| \(\boldsymbol { N }\) | mean | s.d. |
| Low income group | 300 | \(\pounds 6.40\) | \(\pounds 6.69\) |
| High income group | 400 | \(\pounds 7.42\) | \(\pounds 8.13\) |
- Using a \(5 \%\) significance level, test whether or not the two groups differ in the mean amounts spent on tobacco.
- Explain briefly the importance of the central limit theorem in this example.
(2)