| Answer/Working | Marks | Guidance |
|---|---|---|
| Let $D = Y - R$ then $E(D) = -3$; $\text{Var}(D) = 0.8^2 + 1.5^2$ or $1.7^2$ or $2.89$ | B1, M1 | B1 for $E(D) = -3$ (or +3 if using $R - Y$); 1st M1 for $\text{Var}(D) = 0.8^2 + 1.5^2$ o.e. |
| $P(D > 0) = P\left(Z > \frac{0-(-3)}{1.7}\right)$ or $P(Z > 1.7647...)$ | M1 | 2nd M1 for attempt at $P(D > 0)$ must standardise with their -3 and their 1.7 and inequality |
| $= 0.03880655...$ or $1 - 0.9608 = 0.0392$ awrt **0.039** | A1 | A1 for awrt 0.039 |
| **Subtotal: (4 marks)** | | |
| $(R_1 + R_2 + R_3) \sim N\left(45, \sqrt{3 \times 1.5^2}\right)$ ; $4Y \sim N\left(48, \sqrt{4^2 \times 0.8^2}\right)$ | M1A1A1 | 1st M1 for correct mean or variance for either $R_1 + R_2 + R_3$ or $4Y$; 1st A1 for $\left(R_1 + R_2 + R_3\right) \sim N\left(45, \sqrt{6.75}^2\right)$; 2nd A1 for $4Y \sim N\left(48, \sqrt{10.24}^2\right)$ |
| $L = 4Y - (R_1 + R_2 + R_3) \Rightarrow L \sim N\left(3, \sqrt{16.99}\right)$ | M1A1 | 2nd M1 for attempting a suitable $L$ (condone 3R − 4L etc); A1 for correct mean and variance. Sight of $N(\pm 3, 16.99)$ scores 1st 5 marks |
| $P(L > 0) = P\left(Z > \frac{0-3}{\sqrt{16.99}}\right)$ or $P(Z > 0 - 0.7278...)$ [use 0 – 0.73 in tables] | dM1 | 3rd dM1 (dep on 2nd M1) for attempting a prob ($\to$ ans > 0.5) using $\mu_L = \pm 3$ and their $\sigma_L$ |
| $= $ awrt **0.767** | A1 | 4th A1 for awrt 0.767 (Calc: 0.7666384... or tables 0.7673) |
| **Subtotal: (7 marks)** | | |
| $E(X) = 780$ gives $15a + 12b = 780$ | M1A1 | 1st M1 for attempt use $E(X) = 780$ must see a linear equation in $a$ and $b$ using 780; 1st A1 for $15a + 12b = 780$ o.e. e.g. $5a + 4b = 260$ or $a + 0.8b = 52$ etc |
| $[\text{Var}(X)] = 1.5^2 \times a^2 + 0.8^2 \times b^2$ | M1 | 2nd M1 for attempt to find expression for $\text{Var}(X)$ (condone $a$ and $b$ wrong way around) |
| Sub for $a$: $\text{Var}(X) = 2.25(52 - 0.8b)^2 + 0.64 \times b^2$ or $2.08b^2 - 187.2b + 6084$ | M1 | 3rd M1 for forming a quadratic expression for $\text{Var}(X)$ in terms of $a$ or $b$ only (M0 for $k, k \neq 0$) |
| $\frac{d}{db}[\text{Var}(X)] = 0 \Rightarrow 4.16b - 187.2 = 0$ | M1 | 4th M1 suitable method for minimum (e.g. differentiation, or completing square or calc) |
| $\mathbf{b = 45}$ | A1 | 2nd A1 for $b = 45$ |
| So $a = 52 - 0.8 \times 45 = 52 - 36$ | A1 | 3rd A1 for both $b = 45$ and $a = 16$ |
| $\mathbf{a = 16}$ | | Correct answers should be accompanied by evidence for 1st 4 marks |
| **Subtotal: (7 marks)** | | |

**Total: [18 marks]**