S4 June 2005 Q2
2. The standard deviation of the length of a random sample of 8 fence posts produced by a timber yard was 8 mm. A second timber yard produced a random sample of 13 fence posts with a standard deviation of 14 mm.
(a) Test, at the 10% significance level, whether or not there is evidence that the lengths of fence posts produced by these timber yards differ in variability. State your hypotheses clearly. (5)
(b) State an assumption you have made in order to carry out the test in part (a). (1)
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \sigma_1^2 = \sigma_2^2\,;\quad \mathrm{H}_1 : \sigma_1^2 \neq \sigma_2^2\) both | B1 |
| \(\dfrac{s_1^2}{s_2^2} = \dfrac{14^2}{8^2} = 3.0625\) or \(\dfrac{8^2}{14^2} = 0.32653\ldots\) awrt 3.06 or 0.327 | M1 A1 |
| C.V. \(F_{12,7} = 3.57\) C.V.: \(F_{7,12} = \dfrac{1}{3.57} = 0.28011\) | B1 |
| Since 3.0625 is not in the critical region there is insufficient evidence to reject \(\mathrm{H}_0\). There is insufficient evidence of a difference in the variances of the lengths of the fence posts. | A1ft |
| (5) |
| Scheme | Marks |
|---|---|
| The distribution of the population of lengths of fence posts is normally distributed. | B1 |
| (1) |