| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Sampling distribution of mean or linear combination |
| Difficulty | Standard +0.8 This S2 question requires understanding of sampling distributions and probability calculations. Part (a) involves setting up and solving an equation from P(T=18), requiring recognition that this occurs when all three boxes contain 6 eggs. Parts (b) and (c) demand systematic enumeration of all possible outcomes (6,6,6), (6,6,12), etc., calculating their probabilities using the ratio 9:1, then deriving the range distribution. While methodical rather than conceptually deep, it requires careful organization across multiple cases and is more demanding than typical S2 questions. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.02a Discrete probability distributions: general5.02b Expectation and variance: discrete random variables5.02d Binomial: mean np and variance np(1-p) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{n^3}{(n+1)^3} = 0.729 \Rightarrow \frac{n}{n+1} = \sqrt[3]{0.729} \Rightarrow n = 9\) | M1A1cso | M1 for correct equation in \(n\), \(n+1\) and \(0.729\); A1 cso no errors seen. Alt: M1 for \(\frac{9^3}{(9+1)^3} = 0.729\), A1 cso for stating \(n=9\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(T=24) = 0.9^2(1-0.9)\times 3\) | M1 | 1st M1 for either \(p^2(1-p)\times 3\) or \(p(1-p)^2\times 3\) |
| \(P(T=30) = 0.9(1-0.9)^2\times 3\) | M1 | 2nd M1 for \((1-p)^3\) or use of \(1-P(T\neq 36)\) |
| \(P(T=36) = (1-0.9)^3\) | ||
| \(T\): 18, 24, 30, 36 with \(P(T=t)\): 0.729, 0.243, 0.027, 0.001 | A1 A1 | 1st A1 at least 1 correct probability; 2nd A1 all \(t\) values with correct probabilities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(P(R=0) = P(T=18)+P(T=36) = 0.73\) | M1 A1 | M1 correct calculation for either \(P(R=0)\) or \(P(R=6)\); A1 both probabilities correct with correct \(r\) values |
| \(P(R=6) = P(T=24)+P(T=30) = 0.27\) |
## Question 2:
### Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{n^3}{(n+1)^3} = 0.729 \Rightarrow \frac{n}{n+1} = \sqrt[3]{0.729} \Rightarrow n = 9$ | M1A1cso | M1 for correct equation in $n$, $n+1$ and $0.729$; A1 cso no errors seen. Alt: M1 for $\frac{9^3}{(9+1)^3} = 0.729$, A1 cso for stating $n=9$ |
### Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(T=24) = 0.9^2(1-0.9)\times 3$ | M1 | 1st M1 for either $p^2(1-p)\times 3$ or $p(1-p)^2\times 3$ |
| $P(T=30) = 0.9(1-0.9)^2\times 3$ | M1 | 2nd M1 for $(1-p)^3$ or use of $1-P(T\neq 36)$ |
| $P(T=36) = (1-0.9)^3$ | | |
| $T$: 18, 24, 30, 36 with $P(T=t)$: 0.729, 0.243, 0.027, 0.001 | A1 A1 | 1st A1 at least 1 correct probability; 2nd A1 all $t$ values with correct probabilities |
### Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $P(R=0) = P(T=18)+P(T=36) = 0.73$ | M1 A1 | M1 correct calculation for either $P(R=0)$ or $P(R=6)$; A1 both probabilities correct with correct $r$ values |
| $P(R=6) = P(T=24)+P(T=30) = 0.27$ | | |
---2. A farmer sells boxes of eggs.
The eggs are sold in boxes of 6 eggs and boxes of 12 eggs in the ratio $n : 1$
A random sample of three boxes is taken.\\
The number of eggs in the first box is denoted by $X _ { 1 }$\\
The number of eggs in the second box is denoted by $X _ { 2 }$\\
The number of eggs in the third box is denoted by $X _ { 3 }$\\
The random variable $T = X _ { 1 } + X _ { 2 } + X _ { 3 }$\\
Given that $\mathrm { P } ( T = 18 ) = 0.729$
\begin{enumerate}[label=(\alph*)]
\item show that $n = 9$
\item find the sampling distribution of $T$
The random variable $R$ is the range of $X _ { 1 } , X _ { 2 } , X _ { 3 }$
\item Using your answer to part (b), or otherwise, find the sampling distribution of $R$\\
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\hfill \mbox{\textit{Edexcel S2 2018 Q2 [8]}}