S4 June 2009 Q4
4. A farmer set up a trial to assess whether adding water to dry feed increases the milk yield of his cows. He randomly selected 22 cows. Thirteen of the cows were given dry feed and the other 9 cows were given the feed with water added. The milk yields, in litres per day, were recorded with the following results.
| Sample size | Mean | \(s^2\) | |
|---|---|---|---|
| Dry feed | 13 | 25.54 | 2.45 |
| Feed with water added | 9 | 27.94 | 1.02 |
You may assume that the milk yield from cows given the dry feed and the milk yield from cows given the feed with water added are from independent normal distributions.
(a) Test, at the 10% level of significance, whether or not the variances of the populations from which the samples are drawn are the same. State your hypotheses clearly. (5)
(b) Calculate a 95% confidence interval for the difference between the two mean milk yields. (7)
(c) Explain the importance of the test in part (a) to the calculation in part (b). (2)
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \sigma_A^2 = \sigma_B^2,\ \mathrm{H}_1 : \sigma_A^2 \neq \sigma_B^2\) | B1 |
| critical values \(F_{12,8} = 3.28\) and \(\dfrac{1}{F_{8,12}} = 0.35\) | B1 |
| \(\dfrac{s_B^2}{s_A^2} = 2.40 \quad \left(\dfrac{s_A^2}{s_B^2} = 0.416\right)\) | M1A1 |
| Since 2.40 (0.416) is not in the critical region we accept \(\mathrm{H}_0\) and conclude there is no evidence that the two variances are different. | A1ft |
| (5) |
| Scheme | Marks |
|---|---|
| \(S_p^2 = \dfrac{8 \times 1.02 + 12 \times 2.45}{20}\) | M1 |
| \(= 1.878\) | A1 |
| \((27.94 - 25.54) \pm 2.086 \times \sqrt{1.878} \times \sqrt{\dfrac{1}{9} + \dfrac{1}{13}}\) | B1M1 A1ft |
| \((1.16,\ 3.64)\) | A1 A1 |
| (7) |
| Scheme | Marks |
|---|---|
| To calculate the confidence interval the variances need to be equal. | B1 |
| In part (a) the test showed they are equal. | B1 |
| (2) | |
| (14 marks) |