S4 June 2005 Q4
4. A farmer set up a trial to assess the effect of two different diets on the increase in the weight of his lambs. He randomly selected 20 lambs. Ten of the lambs were given diet \(A\) and the other 10 lambs were given diet \(B\). The gain in weight, in kg, of each lamb over the period of the trial was recorded.
(a) State why a paired \(t\)-test is not suitable for use with these data. (1)
(b) Suggest an alternative method for selecting the sample which would make the use of a paired \(t\)-test valid. (1)
(c) Suggest two other factors that the farmer might consider when selecting the sample. (2)
The following paired data were collected.
| Diet \(A\) | 5 | 6 | 7 | 4.6 | 6.1 | 5.7 | 6.2 | 7.4 | 5 | 3 |
|---|---|---|---|---|---|---|---|---|---|---|
| Diet \(B\) | 7 | 7.2 | 8 | 6.4 | 5.1 | 7.9 | 8.2 | 6.2 | 6.1 | 5.8 |
(d) Using a paired \(t\)-test, at the 5% significance level, test whether or not there is evidence of a difference in the weight gained by the lambs using diet \(A\) compared with those using diet \(B\). (8)
(e) State, giving a reason, which diet you would recommend the farmer to use for his lambs. (1)
| Scheme | Marks |
|---|---|
| The data were not collected in pairs. | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| Use data from twin lambs. | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| Age, weight, gender Any two sensible factors. | B1; B1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(d = B - A\) \(d:\ 2,\ 1.2,\ 1,\ 1.8,\ -1,\ 2.2,\ 2,\ -1.2,\ 1.1,\ 2.8\) | M1 |
| \(\sum d = 11.9\,;\quad \sum d^2 = 30.01\) \(\therefore\ \bar{d} = 1.19\,;\quad s^2 = 1.761 \quad (s = 1.327)\) | A1; A1 |
| \(\mathrm{H}_0 : \delta = 0\,;\quad \mathrm{H}_1 : \delta \neq 0\) Allow \(\mu_D\) for \(\delta\) both | B1 |
| \(t = \dfrac{1.19 - 0}{\sqrt{\dfrac{1.761}{10}}} = 2.83574\ldots\) | M1 |
| awrt 2.84 | A1 |
| \(\nu = 9\,;\quad\) CV: \(t = 2.262\) | B1 |
| Since \(2.8357\ldots\) is in the critical region \((t \gt 2.262)\) there is evidence to reject \(\mathrm{H}_0\). The (mean) weight gained by the lambs is different for each diet. | A1ft |
| (8) |
Notes
Alternative (d)
| Scheme | Marks |
|---|---|
| Using non-paired \(t\)-test: \(\mathrm{H}_0 : \mu_A = \mu_B\,;\quad \mathrm{H}_1 : \mu_A \neq \mu_B\) | B1 |
| \(t = \dfrac{\mu_A - \mu_B}{\sqrt{s_p^2\left(\frac{1}{10} + \frac{1}{10}\right)}} = -2.30\) awrt \(-2.30\) | B1 |
| CV: \(|t| = 2.101\) | B1 |
| Conclusion: Mean weight gained is different | B1 |
| NB \(\mu_A = 5.6 \quad \mu_B = 6.79 \quad s_p^2 = 1.342722\ldots\) | |
| (4) |
| Scheme | Marks |
|---|---|
| Diet \(B\) – it has the higher mean | B1 |
| (1) |