3 A large department in a university wished to compare the standards of literacy and numeracy of its students. A random sample of 24 students was taken and sub-divided, randomly, into two groups of 12 . The students in one group took a literacy assessment (scores denoted by \(x\) ); the students in the other group took a numeracy assessment (scores denoted by \(y\) ). The two assessments were designed to give the same distributions of scores when taken by random samples from the general population.
The scores obtained by the students on the two assessments are shown in the table.
| \(x\) | 23 | 42 | 43 | 46 | 48 | 48 | 50 | 54 | 58 | 59 | 62 | 65 |
| \(y\) | 44 | 36 | 63 | 55 | 53 | 58 | 63 | 80 | 61 | 57 | 83 | 54 |
$$\sum x = 598 \quad \sum x ^ { 2 } = 31196 \quad \sum y = 707 \quad \sum y ^ { 2 } = 43543$$
- Carry out an appropriate \(t\) test, at the \(5 \%\) level of significance, to compare the standards of literacy and numeracy.
- State the distributional assumptions required for the \(t\) test to be valid.
Name the test that you would use if the assumptions required for the \(t\) test are thought not to hold. State the hypotheses for this new test.
Explain, in general terms, which of the two tests is more powerful, and why.
A statistician at the university looked at the data and commented that a paired sample design would have been better.
- Explain how a paired sample design would be applied in this context, and how the data would be analysed. Explain also why it would be better than the design used.