S4 June 2018 Q2
2. Jeremiah currently uses a Fruity model of juicer. He agrees to trial a new model of juicer, Zesty. The amounts of juice extracted, \(x\) ml, from each of 9 randomly selected oranges, using the Zesty are summarised as
\[\sum x = 468 \qquad \sum x^2 = 24\,560\]Given that the amounts of juice extracted follow a normal distribution,
Jeremiah knows that, for his Fruity, the mean amount of juice extracted from an orange is 38 ml and the standard deviation of juice extracted from an orange is 5 ml.
He decides that he will replace his Fruity with a Zesty if both
- the mean for the Zesty is more than 20% higher than the mean for his Fruity
and - the standard deviation for the Zesty is less than 5.5 ml.
| Scheme | Marks |
|---|---|
| \(\bar{x} = \dfrac{468}{9} = 52 \qquad s^2 = \dfrac{9}{8}\left(\dfrac{24560}{9} - 52^2\right) = 28\) | M1A1 |
| i) \(t_8 = 2.306\) | B1 |
| 95% CI \(= 52 \pm 2.306 \times \dfrac{\sqrt{\text{"28"}}}{\sqrt{9}}\) | M1 |
| \(= (47.93\ldots,\ 56.06\ldots)\) | A1 |
| ii) 95% CI is given by \(\dfrac{8 \times 28}{17.535} \lt \sigma^2 \lt \dfrac{8 \times 28}{2.180}\) | M1 B1 |
| \(12.77 \lt \sigma^2 \lt 102.75\) | |
| \(3.57 \lt \sigma \lt 10.14\) | M1d A1 |
| (9) |
Notes
M1 attempting \(s\) or \(s^2\)
A1 28 only
(i) B1 CV awrt 2.306
M1 \(\bar{x} \pm t\text{-value} \times \dfrac{\sqrt{\text{their Var}}}{\sqrt{9}}\)
A1 awrt 47.9 and awrt 56.1
(ii) 1st M1 for \(\dfrac{8\,(\text{their } s)^2}{\chi^2}\)
B1 awrt 17.535 & awrt 2.18
M1d Dept on previous M mark. Rearranging leading to interval for \(\sigma\) – must square root
A1 awrt 3.57 and 10.1
| Scheme | Marks |
|---|---|
| \(38 \times 1.2 = 45.6\) or 26% or 1.26 | B1 |
| 45.6 is below the CI for Zesty therefore there is evidence that the mean for Zesty is more than 20% higher than his Fruity | M1 |
| 5.5 is in the CI therefore there is no evidence that the standard deviation is less than 5.5 | M1 |
| He should not change to Zesty | A1cso |
| (4) | |
| (13 marks) |
Notes
B1 45.6 seen or awrt 26% or awrt 1.26
M1 correct reason for mean ft their CI
M1 correct reason for sd ft their CI
A1cso correct conclusion. Not ft on incorrect intervals
(corrected from the printed mark scheme: the scheme prints “below the CI for Fruity”; the interval is the one for the Zesty mean from part (a)(i))