S3 June 2014 (R) Q3
3. A company produces two types of milk powder, ‘Semi-Skimmed’ and ‘Full Cream’. In tests, each type of milk powder is used to make a large number of cups of coffee. The mass, \(S\) grams, of ‘Semi-Skimmed’ milk powder used in one cup of coffee is modelled by \(S \sim \mathrm{N}(4.9, 0.8^2)\). The mass, \(C\) grams, of ‘Full Cream’ milk powder used in one cup of coffee is modelled by \(C \sim \mathrm{N}(2.5, 0.4^2)\)
| Scheme | Marks |
|---|---|
| \(\mathrm{P}(S \gt 2C) = \mathrm{P}(S - 2C \gt 0)\) | B1 |
| \(\mathrm{E}[S - 2C] = 4.9 - 2 \times 2.5 = -0.1\) | M1A1 |
| \(\mathrm{Var}(S - 2C) = 0.64 + 4 \times 0.16 = 1.28\) | M1, M1 |
| \(\mathrm{P}(S - 2C \gt 0) = \mathrm{P}\left(Z \gt \frac{0 - -0.1}{\sqrt{1.28}}\right)\) \(= \mathrm{P}(Z \gt 0.08838\ldots)\) \(= 0.4641\) (tables), or \(0.4648\ldots\) (calculator) accept awrt 0.464 or 0.465 | A1 |
| (6) |
Notes
1st M1 for …\(\pm 4\mathrm{Var}(C)\)
2nd M1 for \(\mathrm{P}(S - 2C \gt 0)\)
3rd M1 ft their expectation and variance but not if \(\mathrm{Var}(S - 2C)\) is negative. (Should lead to \(\mathrm{P}(Z \gt +\text{ve})\))
| Scheme | Marks |
|---|---|
| Let \(T = S_1 + S_2 + \ldots + S_{100}\) | |
| \(\mathrm{E}[T] = 100 \times 4.9 = 490\) | M1A1 |
| \(\mathrm{Var}(T) = 100 \times 0.64 = 64\) | A1 |
| \(\mathrm{P}(T \lt 500) = \mathrm{P}\left(Z \lt \frac{500 - 490}{\sqrt{64}}\right)\) | M1 |
| \(= \mathrm{P}(Z \lt 1.25)\) \(= 0.8944\) | A1 |
| (5) | |
| (11 marks) |
Notes
1st M1 for attempt to find mean or variance of total
1st A1 either correct
2nd A1 both correct
2nd M1 for standardising using 500, their mean and their sd leading to \(\mathrm{P}(Z \lt +\text{ve})\) o.e.