| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2020 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Two or more different variables |
| Difficulty | Standard +0.8 This question requires understanding that differences and linear combinations of independent normal variables are also normal, then correctly identifying parameters (variance of difference, mean/variance of weighted sum). While the calculations are standard once set up, the conceptual step of forming M - J ~ N(170, 220) and 0.2M + 0.5J ~ N(181, 25.6) requires solid understanding beyond routine application, placing it moderately above average difficulty. |
| Spec | 5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(S > L + 200) = P(S - L > 200)\); \(E(S-L) = 380 - 210\ (=170)\) or \(E(S-L-200) = 380-210-200\ (=-30)\) | B1 | These may be implied by next line |
| \(\text{Var}(S-L) = 140+80\ (=220)\) or \(\text{Var}(S-L-200) = 140+80\ (=220)\) | B1 | |
| \(\frac{200 - \text{"170"}}{\sqrt{\text{"220"}}}\) or \(\frac{0 - \text{"}-30\text{"}}{{\sqrt{\text{"220"}}}}\ (= 2.023)\) | M1 | Standardising with *their* values (must be from a combination attempt). Allow with attempted continuity correction |
| \(1 - \phi(\text{"2.023"})\) | M1 | Area consistent with their values |
| \(= 0.0216\) (3sf) | A1 | (0.0234 with continuity correction) |
| 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(E(\text{total cost}) = 380 \times 20 + 210 \times 50\ (= 18100)\) | B1 | or \$181 |
| \(\text{Var}(\text{total cost}) = 140 \times 20^2 + 80 \times 50^2\ (= 256000)\) | B1 | or 25.6 (dollar²). These may be implied by next line |
| \(\frac{19000 - \text{"18100"}}{\sqrt{\text{"256000"}}}\) or \(\frac{190 - \text{"181"}}{\sqrt{\text{"25.6"}}}\ (= 1.778)\) | M1 | Standardising with *their* values (must be from a combination attempt). No mixed methods. Allow with attempted continuity correction |
| \(\phi(\text{"1.778"})\) | M1 | Area consistent with *their* values |
| \(= 0.962\) or \(0.963\) (3sf) | A1 | (0.953 or 0.954 with continuity correction) |
| 5 |
## Question 7(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(S > L + 200) = P(S - L > 200)$; $E(S-L) = 380 - 210\ (=170)$ or $E(S-L-200) = 380-210-200\ (=-30)$ | **B1** | These may be implied by next line |
| $\text{Var}(S-L) = 140+80\ (=220)$ or $\text{Var}(S-L-200) = 140+80\ (=220)$ | **B1** | |
| $\frac{200 - \text{"170"}}{\sqrt{\text{"220"}}}$ or $\frac{0 - \text{"}-30\text{"}}{{\sqrt{\text{"220"}}}}\ (= 2.023)$ | **M1** | Standardising with *their* values (must be from a combination attempt). Allow with attempted continuity correction |
| $1 - \phi(\text{"2.023"})$ | **M1** | Area consistent with their values |
| $= 0.0216$ (3sf) | **A1** | (0.0234 with continuity correction) |
| | **5** | |
## Question 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $E(\text{total cost}) = 380 \times 20 + 210 \times 50\ (= 18100)$ | **B1** | or \$181 |
| $\text{Var}(\text{total cost}) = 140 \times 20^2 + 80 \times 50^2\ (= 256000)$ | **B1** | or 25.6 (dollar²). These may be implied by next line |
| $\frac{19000 - \text{"18100"}}{\sqrt{\text{"256000"}}}$ or $\frac{190 - \text{"181"}}{\sqrt{\text{"25.6"}}}\ (= 1.778)$ | **M1** | Standardising with *their* values (must be from a combination attempt). No mixed methods. Allow with attempted continuity correction |
| $\phi(\text{"1.778"})$ | **M1** | Area consistent with *their* values |
| $= 0.962$ or $0.963$ (3sf) | **A1** | (0.953 or 0.954 with continuity correction) |
| | **5** | |7 Before a certain type of book is published it is checked for errors, which are then corrected. For costing purposes each error is classified as either minor or major. The numbers of minor and major errors in a book are modelled by the independent distributions $\mathrm { N } ( 380,140 )$ and $\mathrm { N } ( 210,80 )$ respectively. You should assume that no continuity corrections are needed when using these models.
A book of this type is chosen at random.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that the number of minor errors is at least 200 more than the number of major errors.\\
The costs of correcting a minor error and a major error are 20 cents and 50 cents respectively.
\item Find the probability that the total cost of correcting the errors in the book is less than $\$ 190$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2020 Q7 [10]}}