A2 October 2021 Q6
6. Elsa is collecting information on the wingspan of two different species of butterfly, Ringlet and Meadow Brown. She takes a random sample of each type of butterfly. The wingspans, \(w\) cm, are summarised in the table below. The wingspans of Ringlet and Meadow Brown butterflies each follow normal distributions.
| Number of butterflies | \(\sum w\) | \(\sum w^2\) | |
|---|---|---|---|
| Ringlet | 8 | 410 | 21 032 |
| Meadow Brown | 6 | 294 | 14 426 |
The \(k\)% confidence interval for the variance of the wingspans of Meadow Brown butterflies is (1.194, 48.54)
| Scheme | Marks | AO |
|---|---|---|
| \(\mathrm{H}_0 : \sigma_r^{\,2} = \sigma_{mb}^{\,2} \qquad \mathrm{H}_1 : \sigma_r^{\,2} \neq \sigma_{mb}^{\,2}\) | B1 | 2.5 |
| \(s_r^2 = \dfrac{1}{7}\left(21032 - 8 \times \left(\dfrac{410}{8}\right)^2\right) = 2.7857\ldots\) | M1 A1 | 2.1 1.1b |
| \(s_{mb}^2 = \dfrac{1}{5}\left(14426 - 6 \times \left(\dfrac{294}{6}\right)^2\right) = 4\) | A1 | 1.1b |
| \(\dfrac{s_{mb}^2}{s_r^2} = 1.4358\ldots\) | M1 | 3.4 |
| CV \(\mathrm{F}_{5,7} = 7.46\) | B1 | 1.1b |
| (\(1.4358\ldots \lt 7.46\) so there is) insufficient evidence to suggest the variances of the wingspans are different. | A1 | 2.2b |
| (7) |
Notes
B1: For both hypotheses in terms of \(\sigma\)
M1: Correct method for \(s_r^2\) or \(s_{mb}^2\)
A1: awrt 2.79 (allow \(\tfrac{39}{14}\))
A1: 4 cao
M1: Using correct model for test statistic with correct ratio
B1: Correct CV
A1ft: Drawing a correct inference in context using their CV and their test statistic (dep on both M marks)
| Scheme | Marks | AO |
|---|---|---|
| \(\chi^2_{5,\alpha} = \dfrac{5 \times \text{‘}4\text{’}}{1.194} \qquad \text{or} \qquad \chi^2_{5,\alpha} = \dfrac{5 \times \text{‘}4\text{’}}{48.54}\) | M1 | 3.1b |
| \(\chi^2_{5,\alpha} = 16.75 \rightarrow \alpha = 0.005 \qquad \text{or} \qquad \chi^2_{5,\alpha} = 0.412 \rightarrow \alpha = 0.995\) | M1 | 1.1b |
| \(k = 99\) | A1 | 1.1b |
| (3) |
Notes
M1: Either correct attempt at \(\chi^2_{5,\alpha}\) with \(\nu = 5\)
M1: Using tables to find appropriate probability
A1: 99 cao
| Scheme | Marks | AO |
|---|---|---|
| \(s_p^{\,2} = \dfrac{7 \times \text{‘}2.7857\ldots\text{’} + 5 \times \text{‘}4\text{’}}{8 + 6 - 2}\ [= 3.29\ldots]\) | M1 | 3.1b |
| \(t_{12} = 2.179\) | B1 | 1.1b |
| \(\left(\tfrac{410}{8} - \tfrac{294}{6}\right) \pm \text{‘}2.179\text{’} \times \sqrt{\text{‘}3.29\text{’}}\sqrt{\tfrac{1}{8} + \tfrac{1}{6}}\) | M1 | 3.4 |
| (awrt 0.115, awrt 4.39) | A1 A1 | 1.1b 1.1b |
| (5) | ||
| (15 marks) |
Notes
M1: Correct expression for \(s_p^{\,2}\)
B1: Correct 95% \(t\)-value
M1: CI in the correct form (may be implied by either A mark)
A1: awrt 0.115
A1: awrt 4.39