S4 June 2006 Q3
3. As part of an investigation into the effectiveness of solar heating, a pair of houses was identified where the mean weekly fuel consumption was the same. One of the houses was then fitted with solar heating and the other was not. Following the fitting of the solar heating, a random sample of 9 weeks was taken and the table below shows the weekly fuel consumption for each house.
| Week | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| Without solar heating | 19 | 19 | 18 | 14 | 6 | 7 | 5 | 31 | 43 |
| With solar heating | 13 | 22 | 11 | 16 | 14 | 1 | 0 | 20 | 38 |
Units of fuel used per week
(a) Stating your hypotheses clearly, test, at the 5% level of significance, whether or not there is evidence that the solar heating reduces the mean weekly fuel consumption. (8)
(b) State an assumption about weekly fuel consumption that is required to carry out this test. (1)
| Scheme | Marks |
|---|---|
| (\(D\) = Without Solar heating − with Solar heating) \(\mathrm{H}_0 : \mu_D = 0 \qquad \mathrm{H}_1 : \mu_D \gt 0\) | B1 |
| \(d:\ 6,\ -3,\ 7,\ -2,\ -8,\ 6,\ 5,\ 11,\ 5\) (Attempt \(d\)’s) | M1 |
| \(\bar{d} = 3, \quad s_d = 6 \quad \left(= \sqrt{\dfrac{369 - 9 \times 3^2}{8}}\right)\) or \(s_d^2 = 36\) \(\left(\dfrac{\Sigma d}{9}\ \text{M1}\right)\) | M1, M1 |
| \(t_8 = \dfrac{3 - 0}{6/\sqrt{9}} = 1.5\) \((\pm)\) | M1 A1 cao |
| \(t_8\) (5% 1 tail c.v.) \(= 1.860\) | B1 |
| Not significant – insufficient evidence (that solar heating has) decreased weekly fuel consumption. | A1ft |
| (8) |
Notes
\(\mu_1 = \mu_2\) etc is B0 \(\left[\bar{D} = 0,\ \bar{D} \gt 0\ \text{B0};\ D = 0,\ D \gt 0\ \text{B0}\right]\)
The scheme links the B1 for the critical value and the final A1ft with an arrow.
| Scheme | Marks |
|---|---|
| Difference in weekly fuel consumption is normally distributed. | B1 |
| (1) | |
| (9 marks) |