| Exam Board | Edexcel |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Sampling distribution theory |
| Difficulty | Moderate -0.8 This is a theory-based question testing definitions and basic concepts (random sample, population, statistic, sampling distribution) with minimal calculation. Parts (a), (b), (d) require recall of definitions, part (c) is immediate application of standardization, and part (e) requires recognizing that a statistic cannot depend on unknown parameters. All parts are straightforward bookwork with no problem-solving or novel insight required, making it easier than average A-level questions. |
| Spec | 2.01a Population and sample: terminology5.05a Sample mean distribution: central limit theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Every possible sample (of size 12 cartons) has an equal chance of being selected. Or sample selected without bias from the dairy/factory. Or sample where all cartons have the same chance/prob of being chosen. | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| [The volumes of] all the cartons of milk. Or the cartons of milk from the dairy/factory. | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(N(0,1)\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| The probability distribution of \(X\) or the distribution of all possible values of \(X\). Or all the values of the statistic and their probabilities. | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Only (II) is not a statistic | B1 | For choosing II only |
| as it contains (unknown) parameters \(\mu\) and/or \(\sigma\). Or it contains unknown parameters. | B1d | Dependent on choosing II only; for correct reason about parameters |
## Question 4:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Every possible sample (of size 12 cartons) has an equal chance of being selected. Or sample selected without bias from the dairy/factory. Or sample where all cartons have the same chance/prob of being chosen. | B1 | |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| [The volumes of] all the cartons of milk. Or the cartons of milk from the dairy/factory. | B1 | |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $N(0,1)$ | B1 | |
### Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| The probability distribution of $X$ or the distribution of all possible values of $X$. Or all the values of the statistic and their probabilities. | B1 | |
### Part (e):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Only (II) is not a statistic | B1 | For choosing II only |
| as it contains (unknown) parameters $\mu$ and/or $\sigma$. Or it contains unknown parameters. | B1d | Dependent on choosing II only; for correct reason about parameters |
---4. The volume of milk, $M$ litres, in cartons produced by a dairy, has distribution $\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)$, where $\mu$ and $\sigma$ are unknown. A random sample of 12 cartons is taken and the volume of milk in each carton is measured ( $M _ { 1 } , M _ { 2 } , \ldots , M _ { 12 }$ ). A statistic $X$ is based on this sample.
\begin{enumerate}[label=(\alph*)]
\item Explain what is meant by "a random sample" in this case.
\item State the population in this case.
\item Write down the distribution of $\frac { M _ { 12 } - \mu } { \sigma }$
\item Explain what you understand by the sampling distribution of $X$.
\item State, giving a reason, which of the following is not a statistic based on this sample.\\
(I) $3 M _ { 1 } + \frac { 2 M _ { 11 } } { 6 }$\\
(II) $\sum _ { i = 1 } ^ { 12 } \left( \frac { M _ { i } - \mu } { \sigma } \right) ^ { 2 }$\\
(III) $\sum _ { i = 1 } ^ { 12 } \left( 2 M _ { i } - 3 \right)$
\end{enumerate}
\hfill \mbox{\textit{Edexcel S2 2018 Q4 [6]}}