A2 June 2024 Q6
6. A researcher set up a trial to assess the effect that a food supplement has on the increase in weight of Herdwick lambs. The researcher randomly selected 8 sets of twin lambs. One of each set of twins was given the food supplement and the other had no food supplement. The gain in weight, in kg, of each lamb over the period of the trial was recorded.
| Set of twin lambs | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | |
|---|---|---|---|---|---|---|---|---|---|
| Weight gain (kg) | With food supplement | 4.1 | 5.3 | 6.0 | 3.6 | 5.9 | 4.2 | 7.1 | 6.4 |
| No food supplement | 5.0 | 4.8 | 5.2 | 3.4 | 5.1 | 3.9 | 7.0 | 6.5 | |
For a pair of twin lambs, the random variable \(W\) represents the weight gain of the lamb given the food supplement minus the weight gain of the lamb not given the food supplement.
Show your working clearly. (5)
The researcher believes that the mean of \(W\) is greater than 200 g
| Scheme | Marks | AO |
|---|---|---|
| The samples are not independent | B1 | 3.5b |
| (1) |
Notes
B1: The idea that samples are not independent. Condone other irrelevant comments provided they do not contradict this.
| Scheme | Marks | AO |
|---|---|---|
| They should consider the birth weight, gender, or whether or not the lambs are premature. oe | B1 B1 | 2.4 2.4 |
| (2) |
Notes
B1: For one suitable comment on twins being identical relating to selecting the sample. Condone start weight. Do not accept age / diets of the lambs
B1: For a second suitable comment.
| Scheme | Marks | AO |
|---|---|---|
| Need the assumption that the underlying distribution of the difference between the weight gains must be normally distributed. | B1 | 2.4 |
| (1) |
Notes
B1: Need the emboldened words.
| Scheme | Marks | AO |
|---|---|---|
| Difference \(-0.9\quad 0.5\quad 0.8\quad 0.2\quad 0.8\quad 0.3\quad 0.1\quad -0.1\) | M1 | 3.1b |
| \(\bar{w} = 0.2125 \qquad s^2 = 0.3041\ldots\quad (s = 0.551\ldots)\) | M1 | 1.1b |
| Confidence interval: \(\text{“}0.2125\text{”} \pm t \times \sqrt{\dfrac{\text{“}0.3041\text{”}}{8}}\) | M1 | 2.1 |
| \(\text{“}0.2125\text{”} \pm 2.998 \times \sqrt{\dfrac{\text{“}0.3041\text{”}}{8}}\) | A1ft | 1.1b |
| \(= (-0.37202\ldots,\ 0.79702\ldots)\) oe | A1 | 1.1b |
| (5) |
Notes
M1: attempting differences (at least 4 correct) implied by awrt 0.304 or 0.551 but not 0.2125
M1: attempt to find \(\bar{w}\) and \(s\) or \(s^2\) for their differences implied by 0.2125 and either awrt 0.304 or 0.551
M1: For using the correct formula their \(\bar{w} \pm t \times \sqrt{\dfrac{\text{their } s^2}{8}}\) where \(|t| \gt 2\) all values need to be substituted in.
A1ft: for their \(\bar{w} \pm\) awrt \(2.998 \times \sqrt{\dfrac{\text{their } s^2}{8}}\) all values need to be substituted in
A1: dependent on all previous method marks (awrt -0.372, awrt 0.797)
SC: If they have carried out a CI for two independent samples allow 3rd M for using their difference of their means and a pooled variance and A1ft using correct formula with awrt 2.624
| Scheme | Marks | AO |
|---|---|---|
| \(\mathrm{H}_0 : \mu_w = 0.2 \qquad \mathrm{H}_1 : \mu_w \gt 0.2\) | B1 | 2.5 |
| 200g = 0.2 kg is in the interval | M1 | 2.1 |
| There is no evidence that \(\mu_w\) is greater than 0.2 oe | A1ft | 2.2b |
| (3) | ||
| (12 marks) |
Notes
B1: For both hypotheses correct in terms of \(\mu\) or \(\mu_w\) Condone 200
M1: For changing 200 g to 0.2 kg (ignore units for this mark) and comparing to their CI
A1ft: Independent of hypotheses. Drawing a correct inference following through on their CI provided 0.2 is within their confidence interval, with no contradictory statements. Does not need to be in context. Accept “insufficient evidence to support the (researcher’s) belief”.