5. An educational researcher is testing the effectiveness of a new method of teaching a topic in mathematics. A random sample of 10 children were taught by the new method and a second random sample of 9 children, of similar age and ability, were taught by the conventional method. At the end of the teaching, the same test was given to both groups of children.
The marks obtained by the two groups are summarised in the table below.
| New method | Conventional method |
| Mean \(( \bar { x } )\) | 82.3 | 78.2 |
| Standard deviation \(( s )\) | 3.5 | 5.7 |
| Number of students \(( n )\) | 10 | 9 |
- Stating your hypotheses clearly and using a \(5 \%\) level of significance, investigate whether or not
- the variance of the marks of children taught by the conventional method is greater than that of children taught by the new method,
- the mean score of children taught by the conventional method is lower than the mean score of those taught by the new method.
[0pt]
[In each case you should give full details of the calculation of the test statistics.]
- State any assumptions you made in order to carry out these tests.
- Find a 95\% confidence interval for the common variance of the marks of the two groups.