| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2023 |
| Session | October |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Sampling distribution of mean or linear combination |
| Difficulty | Standard +0.3 This is a straightforward S2 question requiring basic probability calculations and systematic enumeration. Part (a) is routine expectation calculation, part (b) requires listing combinations (not difficult but time-consuming), and part (c) involves finding medians from the listed combinations—all standard techniques with no novel insight required, making it slightly easier than average. |
| Spec | 5.01a Permutations and combinations: evaluate probabilities5.03c Calculate mean/variance: by integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(0.2\times£10 + 0.3\times£12 + 0.5\times£15\) | M1 | May be implied by a correct answer |
| \(= [£]13.10\) | A1 | Cao. Allow 13.1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| At least 5 possible combinations listed (e.g. {10,10,10}, {10,10,12}×3, {10,10,15}×3, {10,12,15}×6, etc.) | B1 | B1 for at least 5 possible combinations. Ignore repeats |
| All 10 possible combinations listed | B1 | For all 10 possible combinations. Ignore repeats |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(10)=0.2\), \(P(12)=0.3\), \(P(15)=0.5\) | B1 | Correct probabilities |
| Median can be 10, 12 or 15 | B1 | All 3 medians and no extras |
| \(P(M=10) = 0.2^3 + 0.2^2\times0.3\times3 + 0.2^2\times0.5\times3\) or \(1-0.8^3-3\times0.8^2\times0.2\) | M1 | A correct method for one of the probabilities |
| \(P(M=12) = 0.3^3 + 0.3^2\times0.5\times3 + 0.3^2\times0.2\times3 + 0.2\times0.3\times0.5\times6\) | M1 | A correct method for two of the probabilities |
| \(P(M=15) = 0.5^3 + 0.5^2\times0.3\times3 + 0.5^2\times0.2\times3\) or \(1-0.5^3-3\times0.5^2\times0.5\) | M1 | A correct method for all three probabilities, or 3 probabilities that add to 1 |
| \(P(M=10)=\frac{13}{125}=0.104\), \(P(M=12)=\frac{99}{250}=0.396\), \(P(M=15)=\frac{1}{2}=0.5\) | A1 | Cao. Need not be in a table but probabilities must be attached to correct median |
# Question 4:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| $0.2\times£10 + 0.3\times£12 + 0.5\times£15$ | M1 | May be implied by a correct answer |
| $= [£]13.10$ | A1 | Cao. Allow 13.1 |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| At least 5 possible combinations listed (e.g. {10,10,10}, {10,10,12}×3, {10,10,15}×3, {10,12,15}×6, etc.) | B1 | B1 for at least 5 possible combinations. Ignore repeats |
| All 10 possible combinations listed | B1 | For all 10 possible combinations. Ignore repeats |
## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(10)=0.2$, $P(12)=0.3$, $P(15)=0.5$ | B1 | Correct probabilities |
| Median can be 10, 12 or 15 | B1 | All 3 medians and no extras |
| $P(M=10) = 0.2^3 + 0.2^2\times0.3\times3 + 0.2^2\times0.5\times3$ or $1-0.8^3-3\times0.8^2\times0.2$ | M1 | A correct method for one of the probabilities |
| $P(M=12) = 0.3^3 + 0.3^2\times0.5\times3 + 0.3^2\times0.2\times3 + 0.2\times0.3\times0.5\times6$ | M1 | A correct method for two of the probabilities |
| $P(M=15) = 0.5^3 + 0.5^2\times0.3\times3 + 0.5^2\times0.2\times3$ or $1-0.5^3-3\times0.5^2\times0.5$ | M1 | A correct method for all three probabilities, or 3 probabilities that add to 1 |
| $P(M=10)=\frac{13}{125}=0.104$, $P(M=12)=\frac{99}{250}=0.396$, $P(M=15)=\frac{1}{2}=0.5$ | A1 | Cao. Need not be in a table but probabilities must be attached to correct median |
---\begin{enumerate}
\item A manufacturer makes t -shirts in 3 sizes, small, medium and large.
\end{enumerate}
20\% of the t -shirts made by the manufacturer are small and sell for $\pounds 10 30 \%$ of the t -shirts made by the manufacturer are medium and sell for $\pounds 12$
The rest of the t -shirts made by the manufacturer are large and sell for $\pounds 15$\\
(a) Find the mean value of the t -shirts made by the manufacturer.
A random sample of 3 t -shirts made by the manufacturer is taken.\\
(b) List all the possible combinations of the individual selling prices of these 3 t-shirts.\\
(c) Find the sampling distribution of the median selling price of these 3 t-shirts.
\hfill \mbox{\textit{Edexcel S2 2023 Q4 [10]}}