| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2023 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Calculating bias of estimator |
| Difficulty | Moderate -0.8 This is a straightforward S3 question testing standard definitions and properties of estimators. Part (a) tests understanding of what a statistic is (must not depend on unknown parameters). Parts (b)-(c) involve routine calculation of E(S) to show bias. Parts (d)-(e) apply standard results about expectation and variance of linear combinations. All steps are mechanical applications of well-rehearsed techniques with no problem-solving or novel insight required. |
| Spec | 5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| It is not a statistic as it involves unknown [population] parameter | B1 | Allow \(\sigma\) is unknown (not "variance is unknown") |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(E(S)=E\!\left(\frac{3}{5}X_1+\frac{5}{7}X_2\right)=\frac{3}{5}E(X_1)+\frac{5}{7}E(X_2)\) | M1 | For writing or using \(E(S)=aE(X_1)+bE(X_2)\); condone missing subscripts |
| \(=\frac{3}{5}\mu+\frac{5}{7}\mu=\frac{46}{35}\mu \neq \mu\), so \(S\) is a biased estimator for \(\mu\) | A1 | cao (allow \(1.31\mu \neq \mu\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{46}{35}\mu - \mu = \frac{11}{35}\mu\) | B1ft | Follow through their part (b) \(-\mu\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(E(Y)=aE(X_1)+bE(X_2)=\mu \Rightarrow (a+b)\mu=\mu\) | M1 | For writing or using \(E(Y)=aE(X_1)+bE(X_2)=\mu\) |
| \(a+b=1\) | A1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\text{Var}(Y)=a^2\text{Var}(X_1)+b^2\text{Var}(X_2)=(a^2+b^2)\sigma^2\) | M1 | For writing or using \(\text{Var}(Y)=a^2\text{Var}(X_1)+b^2\text{Var}(X_2)\) |
| \(\text{Var}(Y)=\left(a^2+(1-a)^2\right)\sigma^2\) | M1 | For substitution of \(b=1-a\) ft their part (d) |
| \(\text{Var}(Y)=(2a^2-2a+1)\sigma^2\) | A1* | Answer given so no incorrect working must be seen |
## Question 3:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| It is not a statistic as it involves unknown [population] parameter | B1 | Allow $\sigma$ is unknown (not "variance is unknown") |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $E(S)=E\!\left(\frac{3}{5}X_1+\frac{5}{7}X_2\right)=\frac{3}{5}E(X_1)+\frac{5}{7}E(X_2)$ | M1 | For writing or using $E(S)=aE(X_1)+bE(X_2)$; condone missing subscripts |
| $=\frac{3}{5}\mu+\frac{5}{7}\mu=\frac{46}{35}\mu \neq \mu$, so $S$ is a biased estimator for $\mu$ | A1 | cao (allow $1.31\mu \neq \mu$) |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{46}{35}\mu - \mu = \frac{11}{35}\mu$ | B1ft | Follow through their part (b) $-\mu$ |
### Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $E(Y)=aE(X_1)+bE(X_2)=\mu \Rightarrow (a+b)\mu=\mu$ | M1 | For writing or using $E(Y)=aE(X_1)+bE(X_2)=\mu$ |
| $a+b=1$ | A1 | cao |
### Part (e):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\text{Var}(Y)=a^2\text{Var}(X_1)+b^2\text{Var}(X_2)=(a^2+b^2)\sigma^2$ | M1 | For writing or using $\text{Var}(Y)=a^2\text{Var}(X_1)+b^2\text{Var}(X_2)$ |
| $\text{Var}(Y)=\left(a^2+(1-a)^2\right)\sigma^2$ | M1 | For substitution of $b=1-a$ ft their part (d) |
| $\text{Var}(Y)=(2a^2-2a+1)\sigma^2$ | A1* | Answer given so no incorrect working must be seen |
---\begin{enumerate}
\item A random sample of 2 observations, $X _ { 1 }$ and $X _ { 2 }$, is taken from a population with unknown mean $\mu$ and unknown variance $\sigma ^ { 2 }$\\
(a) Explain why $\frac { X _ { 1 } - X _ { 2 } } { \sigma }$ is not a statistic.
\end{enumerate}
$$S = \frac { 3 } { 5 } X _ { 1 } + \frac { 5 } { 7 } X _ { 2 }$$
(b) Show that $S$ is a biased estimator of $\mu$\\
(c) Hence find the bias, in terms of $\mu$, when $S$ is used as an estimator of $\mu$
Given that $Y = a X _ { 1 } + b X _ { 2 }$ is an unbiased estimator of $\mu$, where $a$ and $b$ are constants,\\
(d) find an equation, in terms of $a$ and $b$, that must be satisfied.\\
(e) Using your answer to part (d), show that $\operatorname { Var } ( Y ) = \left( 2 a ^ { 2 } - 2 a + 1 \right) \sigma ^ { 2 }$
\hfill \mbox{\textit{Edexcel S3 2023 Q3 [9]}}