| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Finding unbiased estimator constraints |
| Difficulty | Standard +0.8 This S4 question requires understanding of unbiased estimators and variance calculations for combined samples. While the algebraic manipulation is straightforward, it demands careful application of expectation and variance properties across multiple parts, with the final part requiring interpretation and comparison of estimators. The multi-step nature and need to work with abstract parameters (rather than numerical computation) places it above average difficulty. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.05b Unbiased estimates: of population mean and variance |
A bag contains marbles of which an unknown proportion $p$ is red. A random sample of $n$ marbles is drawn, with replacement, from the bag. The number $X$ of red marbles drawn is noted.
A second random sample of $m$ marbles is drawn, with replacement. The number $Y$ of red marbles drawn is noted.
Given that $p_1 = \frac{aX}{n} + \frac{bY}{m}$ is an unbiased estimator of $p_1$,
\begin{enumerate}[label=(\alph*)]
\item show that $a + b = 1$. [4]
\end{enumerate}
Given that $p_2 = \frac{(X + Y)}{n + m}$
\begin{enumerate}[label=(\alph*)]
\item[(b)] show that $p_2$ is an unbiased estimator for $p$. [3]
\item Show that the variance of $p_1$ is p(1 - $p$)$\left(\frac{a^2}{n} + \frac{b^2}{m}\right)$. [3]
\item Find the variance of $p_2$. [3]
\item Given that $a = 0.4$, $m = 10$ and $n = 20$, explain which estimator $p_1$ or $p_2$ you should use. [4]
\end{enumerate}
(Total 17 marks)
\hfill \mbox{\textit{Edexcel S4 Q7 [17]}}