S4 June 2007 Q3
3. The lengths, \(x\) mm, of the forewings of a random sample of male and female adult butterflies are measured. The following statistics are obtained from the data.
| No. of butterflies | Sample mean \(\bar{x}\) | \(\sum x^2\) | |
|---|---|---|---|
| Females | 7 | 50.6 | 17 956.5 |
| Males | 10 | 53.2 | 28 335.1 |
(a) Assuming the lengths of the forewings are normally distributed test, at the 10% level of significance, whether or not the variances of the two distributions are the same. State your hypotheses clearly. (7)
(b) Stating your hypotheses clearly test, at the 5% level of significance, whether the mean length of the forewings of the female butterflies is less than the mean length of the forewings of the male butterflies. (6)
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \sigma^2_{\mathrm{F}} = \sigma^2_{\mathrm{M}} \qquad \mathrm{H}_1 : \sigma^2_{\mathrm{F}} \neq \sigma^2_{\mathrm{M}}\) | B1 |
| \(s^2_{\mathrm{F}} = \dfrac{1}{6}(17956.5 - 7 \times 50.6^2) = \dfrac{33.98}{6} = 5.66333\ldots\) | B1 |
| \(s^2_{\mathrm{M}} = \dfrac{1}{9}(28335.1 - 10 \times 53.2^2) = \dfrac{32.7}{9} = 3.63333\ldots\) | B1 |
| \(\dfrac{s^2_{\mathrm{F}}}{s^2_{\mathrm{M}}} = 1.5587\ldots\) (Reciprocal 0.6415) | M1 A1 |
| \(F_{6,9} = 3.37\) (or 0.24) | B1 |
| Not in critical region. Variances of the two distributions are the same | A1 |
| (7) |
Notes
Need to have variance and the same o.e.
| Scheme | Marks |
|---|---|
| \(\mathrm{H}_0 : \mu_{\mathrm{F}} = \mu_{\mathrm{M}} \qquad \mathrm{H}_1 : \mu_{\mathrm{F}} \lt \mu_{\mathrm{M}}\) | B1 |
| Pooled estimate \(s^2 = \dfrac{6 \times 5.66333\ldots + 9 \times 3.63333}{15}\) \(= 4.44533\) \(s = 2.11\) | M1 |
| \(t = \dfrac{50.6 - 53.2}{2.11\sqrt{\dfrac{1}{7} + \dfrac{1}{10}}} = \pm 2.50\) | M1 A1 |
| C.V. \(t_{15}(5\%) = \pm 1.753\) | B1 |
| Significant. The mean length of the females forewing is less than the length of the males forewing | A1 |
| (6) |
Notes
Need female and forewing (wing)