S4 June 2013 (R) Q5
5. Students studying for their Mathematics GCSE are assessed by two examination papers. A teacher believes that on average the score on paper I is more than 1 mark higher than the score on paper II. To test this belief the scores of 8 randomly selected students are recorded. The results are given in the table below.
| Student | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) |
| Score on paper I | 57 | 63 | 68 | 81 | 43 | 65 | 52 | 31 |
| Score on paper II | 53 | 62 | 61 | 78 | 44 | 64 | 43 | 29 |
Assuming that the scores are normally distributed and stating your hypotheses clearly, test at the 5% level of significance whether or not there is evidence to support the teacher’s belief. (8)
| Scheme | Marks |
|---|---|
| D = Paper I score − paper II score | |
| \(\mathrm{H}_0 : \mu_\mathrm{D} = 1 \quad \mathrm{H}_1 : \mu_\mathrm{D} \gt 1\) | B1 |
| \(d\): 4, 1, 7, 3, −1, 1, 9, 2 | M1 |
| \(\bar{d} = 3.25\); \(s^2 = \dfrac{162 - 8 \times 3.25^2}{7} = 11.07\ldots\) \((s = 3.32)\) | M1;M1 |
| \(t_7 = \dfrac{3.25 - 1}{3.32/\sqrt{8}} = 1.9126\ldots\) awrt 1.91 | M1A1 |
| \(t_7(5\%) = 1.895\) | B1 |
| There is evidence to support the teacher’s belief or the score on paper I is more than one mark higher than on paper II | A1 ft |
| (8) | |
| (8 marks) |
Notes
1st M1 for attempting differences
2nd M1 for attempting \(\bar{d}\)
3rd M1 for attempting \(s_d\) or \({s_d}^2\), correct expression with their \(\sum d^2\) and \(\bar{d}\) or correct calculation (to 2 sf or better)
4th M1 for use of \(\dfrac{\bar{d} - 1}{s/\sqrt{8}}\), ft their values.
1st A1 awrt 1.91
2nd B1 for 1.895
2nd A1 contextual conclusion ft their values.
SC if they use a 2 sample test they may get the first B1 for \(\mathrm{H}_0 : \mu_\mathrm{I} - \mu_\mathrm{II} = 1\) and \(\mathrm{H}_1 : \mu_\mathrm{I} - \mu_\mathrm{II} \gt 1\)