| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2022 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Sum versus sum comparison |
| Difficulty | Standard +0.3 This question tests standard application of linear combinations of normal variables with clear formulas. Part (a) requires forming L - 4S and finding P(L - 4S > 0), while part (b) needs 12L + 25S and comparing to 1000. Both are routine calculations once the correct combination is identified, with no novel insight required beyond applying the standard results for sums/differences of independent normals. |
| Spec | 5.04b Linear combinations: of normal distributions5.05a Sample mean distribution: central limit theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(E(D) = 53 - (4 \times 14) = -3\) | B1 | OE Give at early stage |
| \(\text{Var}(D) = 11 + 4^2 \times 3 \ [= 59]\) | B1 | or \(\sqrt{(11 + 4^2 \times 3)}\) \((= 7.68\) (3 s.f.)); Give at early stage |
| \(\frac{0-(-3)}{\sqrt{59}} \ [= 0.391]\) | M1 | For standardising with *their* values (var must be from a combination attempt); Ignore continuity correction attempts |
| \(1 - \Phi(\text{`}0.391\text{'})\) | M1 | For area consistent with *their* values |
| \(0.348\) (3 s.f.) | A1 | As final answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(E(T) = 12 \times 53 + 25 \times 14 \ [= 986]\) | B1 | Give at early stage (N.B. accept \(E(T-1000) = -14\)) |
| \(\text{Var}(T) = 12 \times 11 + 25 \times 3 \ [= 207]\) | B1 | Or \(\sqrt{(12\times11 + 25\times3)}\) \((= 14.4\) (3sf)); Give at early stage |
| \(\frac{1000 - 986}{\sqrt{207}} \ [= 0.973]\) | M1 | For standardising with *their* values (var must be from a combination attempt); Ignore continuity correction attempts |
| \(\Phi(\text{`}0.973\text{'})\) | M1 | For area consistent with *their* values |
| \(0.835\) (3 sf) | A1 | As final answer |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $E(D) = 53 - (4 \times 14) = -3$ | **B1** | OE Give at early stage |
| $\text{Var}(D) = 11 + 4^2 \times 3 \ [= 59]$ | **B1** | or $\sqrt{(11 + 4^2 \times 3)}$ $(= 7.68$ (3 s.f.)); Give at early stage |
| $\frac{0-(-3)}{\sqrt{59}} \ [= 0.391]$ | **M1** | For standardising with *their* values (var must be from a combination attempt); Ignore continuity correction attempts |
| $1 - \Phi(\text{`}0.391\text{'})$ | **M1** | For area consistent with *their* values |
| $0.348$ (3 s.f.) | **A1** | As final answer |
---
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $E(T) = 12 \times 53 + 25 \times 14 \ [= 986]$ | **B1** | Give at early stage (N.B. accept $E(T-1000) = -14$) |
| $\text{Var}(T) = 12 \times 11 + 25 \times 3 \ [= 207]$ | **B1** | Or $\sqrt{(12\times11 + 25\times3)}$ $(= 14.4$ (3sf)); Give at early stage |
| $\frac{1000 - 986}{\sqrt{207}} \ [= 0.973]$ | **M1** | For standardising with *their* values (var must be from a combination attempt); Ignore continuity correction attempts |
| $\Phi(\text{`}0.973\text{'})$ | **M1** | For area consistent with *their* values |
| $0.835$ (3 sf) | **A1** | As final answer |
---6 The masses, in kilograms, of large and small sacks of grain have the distributions $\mathrm { N } ( 53,11 )$ and $\mathrm { N } ( 14,3 )$ respectively.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that the mass of a randomly chosen large sack is greater than four times the mass of a randomly chosen small sack.
\item A lift can safely carry a maximum mass of 1000 kg .
Find the probability that the lift can safely carry 12 randomly chosen large sacks and 25 randomly chosen small sacks.\\
$7 X$ is a random variable with distribution $\operatorname { Po } ( 2.90 )$. A random sample of 100 values of $X$ is taken. Find the probability that the sample mean is less than 2.88 .\\
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 2022 Q6 [10]}}