5. Mr Alan and Ms Burns are two Mathematics teachers teaching mixed ability groups of students in a large college. At the end of the college year all students took the same examination. A random sample of 29 of Mr Alan's students and a random sample of 26 of Ms Burns' students are chosen. The results are summarised in the table below.
| Sample Size, \(n\) | Mean, \(\bar { x }\) | Standard Deviation, \(s\) |
| Mr Alan | 29 | 80 | 10 |
| Ms Burns | 26 | 74 | 15 |
- Stating your hypotheses clearly, test, at the \(10 \%\) level of significance whether there is evidence that there is a difference in the mean scores of their students.
Ms Burns thinks the comparison was unfair as the examination was set by Mr Alan. She looks up a different set of examination results for these students and, although Mr Alan's sample has a higher mean, she calculates the test statistic for this new set of results to be 1.6
However, Mr Alan now claims that the mean marks of his students are higher than the mean marks of Ms Burns' students.
- Test Mr Alan's claim, stating the hypotheses and critical values you would use. Use a \(10 \%\) level of significance.