| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2014 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Mixed sum threshold probability |
| Difficulty | Standard +0.8 This question requires understanding of linear combinations of normal distributions and forming appropriate combinations (4A + 6B for part i, A - 0.9B for part ii). While the mechanics are standard S2 content, students must correctly identify the required distributions, handle the unit conversion in part (i), and recognize the less obvious 90% comparison structure in part (ii), making it moderately challenging. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Part (i) | ||
| \(4 \times 125 + 6 \times 130 (= 1280)\) | B1 | Give at early stage. Could be implied by 220. (If B0B0 then 1.28 and 0.009744 can score B1B1). |
| \(4 \times 30^2 + 6 \times 32^2 (= 9744)\) | B1 | |
| \((\pm)\frac{1500-1280}{\sqrt{9744}}(= 2.229)\) | M1 | Standardising. Accept sd/var mix. Must be from combination attempt. |
| \(\Phi('2.229')\) | M1 | Use of tables and correct area consistent with their working |
| \(= 0.987\) (3 sf) | A1 | [5] cwo |
| Part (ii) | ||
| \(125 - 0.9(130) (= 8)\) (or \(-8\)) | B1 | Give at early stage. (If B0B0 scored then accept 0.008 and 0.0017944 for B1B1) |
| \(30^2 + 0.9^2(32^2) (= 1729.44)\) | B1 | |
| \((\pm)\frac{0-'8'}{\sqrt{1729.44}}(= -0.192)\) | M1 | Accept sd/var mix. Must come from a linear combination. |
| \(\Phi('0.192')\) | M1 | Use of tables and correct area consistent with their working (unclear M0) |
| \(= 0.576\) (3 sf) | A1 | [5] |
| Total | [10] |
| **Part (i)** |
|---|
| $4 \times 125 + 6 \times 130 (= 1280)$ | B1 | Give at early stage. Could be implied by 220. (If B0B0 then 1.28 and 0.009744 can score B1B1). |
| $4 \times 30^2 + 6 \times 32^2 (= 9744)$ | B1 | |
| $(\pm)\frac{1500-1280}{\sqrt{9744}}(= 2.229)$ | M1 | Standardising. Accept sd/var mix. Must be from combination attempt. |
| $\Phi('2.229')$ | M1 | Use of tables and correct area consistent with their working |
| $= 0.987$ (3 sf) | A1 | [5] cwo |
| **Part (ii)** |
|---|
| $125 - 0.9(130) (= 8)$ (or $-8$) | B1 | Give at early stage. (If B0B0 scored then accept 0.008 and 0.0017944 for B1B1) |
| $30^2 + 0.9^2(32^2) (= 1729.44)$ | B1 | |
| $(\pm)\frac{0-'8'}{\sqrt{1729.44}}(= -0.192)$ | M1 | Accept sd/var mix. Must come from a linear combination. |
| $\Phi('0.192')$ | M1 | Use of tables and correct area consistent with their working (unclear M0) |
| $= 0.576$ (3 sf) | A1 | [5] |
| **Total** | [10] |4 The masses, in grams, of tomatoes of type $A$ and type $B$ have the distributions $\mathrm { N } \left( 125,30 ^ { 2 } \right)$ and $\mathrm { N } \left( 130,32 ^ { 2 } \right)$ respectively.\\
(i) Find the probability that the total mass of 4 randomly chosen tomatoes of type $A$ and 6 randomly chosen tomatoes of type $B$ is less than 1.5 kg .\\
(ii) Find the probability that a randomly chosen tomato of type $A$ has a mass that is at least $90 \%$ of the mass of a randomly chosen tomato of type $B$.
\hfill \mbox{\textit{CAIE S2 2014 Q4 [10]}}