| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2016 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Fixed container with random contents |
| Difficulty | Standard +0.8 This question requires understanding of linear combinations of normal variables (part i: X+Y+20) and forming a new normal variable from a linear inequality (part ii: X > 4.1Y becomes X - 4.1Y > 0). Part (ii) particularly requires insight to recognize that the inequality can be rewritten as a linear combination. While the calculations are standard once set up, the conceptual leap in part (ii) elevates this above routine S2 questions. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(T \sim N(520, 70)\) | B1 | for \(N(520,...)\) or \(N(500,...)\) if standardising with 510 |
| \(\frac{530-520}{\sqrt{70}}\) \((= 1.195)\) | B1 | for Var \(= 70\) seen or implied |
| \(\phi(1.195)\) | M1 | ft their E and Var; allow without \(\sqrt{}\); finding correct area consistent with working |
| \(= 0.884\) (3 sf) | M1, A1 [5] | CWO |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(E(T) = -10\) | B1 | or \(+10\) for \(T < 0\) |
| \(\text{Var}(T) = 50 + 4.1^2 \times 20\) \((= 386.2)\) | B1 | Seen or implied |
| \(\frac{0-(-10)}{\sqrt{386.2}}\) \((= 0.509)\) | M1 | ft their E and Var; allow without \(\sqrt{}\); finding correct area consistent with working |
| \(1 - \phi(0.509)\) | M1 | |
| \(= 0.305\) (3 sf) | A1 [5] | CWO |
## Question 5:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $T \sim N(520, 70)$ | B1 | for $N(520,...)$ or $N(500,...)$ if standardising with 510 |
| $\frac{530-520}{\sqrt{70}}$ $(= 1.195)$ | B1 | for Var $= 70$ seen or implied |
| $\phi(1.195)$ | M1 | ft their E and Var; allow without $\sqrt{}$; finding correct area consistent with working |
| $= 0.884$ (3 sf) | M1, A1 [5] | CWO |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $E(T) = -10$ | B1 | or $+10$ for $T < 0$ |
| $\text{Var}(T) = 50 + 4.1^2 \times 20$ $(= 386.2)$ | B1 | Seen or implied |
| $\frac{0-(-10)}{\sqrt{386.2}}$ $(= 0.509)$ | M1 | ft their E and Var; allow without $\sqrt{}$; finding correct area consistent with working |
| $1 - \phi(0.509)$ | M1 | |
| $= 0.305$ (3 sf) | A1 [5] | CWO |5 Each box of Fruity Flakes contains $X$ grams of flakes and $Y$ grams of fruit, where $X$ and $Y$ are independent random variables, having distributions $\mathrm { N } ( 400,50 )$ and $\mathrm { N } ( 100,20 )$ respectively. The weight of each box, when empty, is exactly 20 grams. A full box of Fruity Flakes is chosen at random.\\
(i) Find the probability that the total weight of the box and its contents is less than 530 grams.\\
(ii) Find the probability that the weight of flakes in the box is more than 4.1 times the weight of fruit in the box.
\hfill \mbox{\textit{CAIE S2 2016 Q5 [10]}}