| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2014 |
| Session | June |
| Marks | 19 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Estimator properties and bias |
| Difficulty | Challenging +1.2 This is a structured multi-part question on estimator properties that follows a standard template for S4. Parts (a)-(b) involve routine integration and variance scaling. Parts (c)-(e) require understanding of consistency and bias but with formulas provided. The most challenging aspect is comparing estimators using variance, but this is a standard S4 technique. The calculations are straightforward once the concepts are understood, making this slightly above average difficulty for Further Maths Statistics. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration |
**Total: 19 marks**
Emily is monitoring the level of pollution in a river. Over a period of time she has found that the amount of pollution, $X$, in a $100$ ml sample of river water has a continuous distribution with probability density function $f(x)$ given by
$$f(x) = \begin{cases} \dfrac{2x}{a^2} & 0 \leq x \leq a \\ 0 & \text{otherwise} \end{cases}$$
where $a$ is a constant.
Emily takes a random sample $X_1, X_2, X_3, \ldots, X_n$ to try to estimate the value of $a$.
**(a) Show that $E(X) = \dfrac{2a}{3}$ and $\text{Var}(X) = \dfrac{a^2}{18n}$**
(4 marks)
The random variable $\hat{S} = pX$, where $p$ is a constant, is an unbiased estimator of $a$.
**(b) Write down the value of $p$ and find $\text{Var}(S)$.**
(2 marks)
Felix suggests using the statistic $M = \max\{X_1, X_2, X_3, \ldots, X_n\}$ as an estimator of $a$.
He calculates $E(M) = \dfrac{2n}{2n+1}a$ and $\text{Var}(M) = \dfrac{n}{(n+1)^2(2n+1)^2}a^2$
**(c) State, giving your reasons, whether or not $M$ is a consistent estimator of $a$.**
(3 marks)
The random variable $T = qM$, where $q$ is a constant, is an unbiased estimator of $a$.
**(d) Write down, in terms of $n$, the value of $q$ and find $\text{Var}(T)$.**
(3 marks)
**(e) State, giving your reasons, which of $S$ or $T$ you would recommend Emily use as an estimator of $a$.**
(3 marks)
Emily took a sample of $5$ values of $X$ and obtained the following:
$$5.3 \quad 4.3 \quad 5.7 \quad 7.8 \quad 6.9$$
**(f) Calculate the estimate of $a$ using your recommended estimator from part (e).**
(2 marks)
**(g) Find the standard error of your estimate, giving your answer to 2 decimal places.**
(2 marks)6. Emily is monitoring the level of pollution in a river. Over a period of time she has found that the amount of pollution, $X$, in a 100 ml sample of river water has a continuous distribution with probability density function $\mathrm { f } ( x )$ given by
$$f ( x ) = \left\{ \begin{array} { c c }
\frac { 2 x } { a ^ { 2 } } & 0 \leqslant x \leqslant a \\
0 & \text { otherwise }
\end{array} \right.$$
where $a$ is a constant.
Emily takes a random sample $X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { n }$ to try to estimate the value of $a$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\mathrm { E } ( \bar { X } ) = \frac { 2 a } { 3 }$ and $\operatorname { Var } ( \bar { X } ) = \frac { a ^ { 2 } } { 18 n }$
The random variable $S = p \bar { X }$, where $p$ is a constant, is an unbiased estimator of $a$.
\item Write down the value of $p$ and find $\operatorname { Var } ( S )$.
Felix suggests using the statistic $M = \max \left\{ X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { n } \right\}$ as an estimator of $a$.\\
He calculates $\mathrm { E } ( M ) = \frac { 2 n } { 2 n + 1 } a$ and $\operatorname { Var } ( M ) = \frac { n } { ( n + 1 ) ( 2 n + 1 ) ^ { 2 } } a ^ { 2 }$
\item State, giving your reasons, whether or not $M$ is a consistent estimator of $a$.
The random variable $T = q M$, where $q$ is a constant, is an unbiased estimator of $a$.
\item Write down, in terms of $n$, the value of $q$ and find $\operatorname { Var } ( T )$.
\item State, giving your reasons, which of $S$ or $T$ you would recommend Emily use as an estimator of $a$.
Emily took a sample of 5 values of $X$ and obtained the following:\\
5.3\\
4.3\\
$\begin{array} { l l } 5.7 & 7.8 \end{array}$\\
6.9
\item Calculate the estimate of $a$ using your recommended estimator from part (e).
\item Find the standard error of your estimate, giving your answer to 2 decimal places.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 2014 Q6 [19]}}