| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Discrete Random Variables |
| Type | Showing estimator is unbiased |
| Difficulty | Moderate -0.3 This is a straightforward theoretical statistics question testing standard properties of estimators. Part (a) requires simple recall of sampling distribution properties, parts (b) and (c) involve routine algebraic manipulation of expectations and variances using linearity, and part (d) requires comparing variances. While it requires careful algebra across multiple parts (11 marks total), all techniques are standard bookwork with no novel insight or problem-solving required—making it slightly easier than a typical A-level question. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.05b Unbiased estimates: of population mean and variance |
The value of orders, in £, made to a firm over the internet has distribution N($\mu, \sigma^2$). A random sample of $n$ orders is taken and $\bar{X}$ denotes the sample mean.
\begin{enumerate}[label=(\alph*)]
\item Write down the mean and variance of $\bar{X}$ in terms of $\mu$ and $\sigma^2$. [2]
\end{enumerate}
A second sample of $m$ orders is taken and $\bar{Y}$ denotes the mean of this sample.
An estimator of the population mean is given by
$$U = \frac{n\bar{X} + m\bar{Y}}{n + m}$$
\begin{enumerate}[label=(\alph*)]
\item[(b)] Show that $U$ is an unbiased estimator for $\mu$. [3]
\item Show that the variance of $U$ is $\frac{\sigma^2}{n + m}$. [4]
\item State which of $\bar{X}$ or $U$ is a better estimator for $\mu$. Give a reason for your answer. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 Q2 [11]}}