| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Statistics vs non-statistics identification |
| Difficulty | Moderate -0.8 This is a straightforward definitional question testing whether students understand that a statistic cannot depend on unknown population parameters (μ, σ). Part (a) requires only recall of the definition, and part (b) is routine application of variance rules for linear combinations. No problem-solving or novel insight required. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.05a Sample mean distribution: central limit theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Only contains known data / function of data only / no population parameters | B1 | First B1 for known/no unknowns |
| Therefore it is a statistic | B1d | Second B1 dependent on first B1 for 'Yes'/is a statistic |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| (ii) and (iii) contain unknown parameters / population parameters \(\mu\) and/or \(\sigma\) | B1 | Third B1 for unknowns in both (ii) and (iii) |
| Therefore it is not a statistic | B1d | Fourth B1 dependent on third B1 for 'No'/not a statistic in both (ii) and (iii) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(E\left(\frac{3X_1 - X_{20}}{2}\right) = \frac{3\mu - \mu}{2} = \mu\) | B1 | B1 for \(\mu\) |
| \(\text{Var}\left(\frac{3X_1 - X_{20}}{2}\right) = \frac{9\sigma^2 + \sigma^2}{2^2}\) | M1 | M1 for some squaring on numerator or denominator and must add on numerator |
| \(= \frac{5\sigma^2}{2}\) | A1 | A1 for \(\frac{5\sigma^2}{2}\) o.e. |
# Question 2:
## Part (a)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| Only contains **known** data / function of data only / no population parameters | B1 | First B1 for known/no unknowns |
| Therefore it **is a statistic** | B1d | Second B1 dependent on first B1 for 'Yes'/is a statistic |
## Part (a)(ii)(iii)
| Answer | Mark | Guidance |
|--------|------|----------|
| (ii) and (iii) contain **unknown** parameters / population parameters $\mu$ and/or $\sigma$ | B1 | Third B1 for unknowns in **both** (ii) and (iii) |
| Therefore it is **not a statistic** | B1d | Fourth B1 dependent on third B1 for 'No'/not a statistic in **both** (ii) and (iii) |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $E\left(\frac{3X_1 - X_{20}}{2}\right) = \frac{3\mu - \mu}{2} = \mu$ | B1 | B1 for $\mu$ |
| $\text{Var}\left(\frac{3X_1 - X_{20}}{2}\right) = \frac{9\sigma^2 + \sigma^2}{2^2}$ | M1 | M1 for some squaring on numerator or denominator **and** must **add** on numerator |
| $= \frac{5\sigma^2}{2}$ | A1 | A1 for $\frac{5\sigma^2}{2}$ o.e. |
---2. The weights of pears in an orchard are assumed to have unknown mean $\mu$ and unknown standard deviation $\sigma$.
A random sample of 20 pears is taken and their weights recorded.\\
The sample is represented by $X _ { 1 } , X _ { 2 } , \ldots , X _ { 20 }$. State whether or not the following are statistics. Give reasons for your answers.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item $\frac { X _ { 1 } + 3 X _ { 20 } } { 2 }$
\item $\sum _ { i = 1 } ^ { 20 } \left( X _ { i } - \mu \right)$
\item $\sum _ { i = 1 } ^ { 20 } \left( \frac { X _ { i } - \mu } { \sigma } \right)$
\end{enumerate}\item Find the mean and variance of $\frac { 3 X _ { 1 } - X _ { 20 } } { 2 }$
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2014 Q2 [7]}}