| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Validity and assumptions questions |
| Difficulty | Standard +0.3 This is a straightforward S4 question testing standard definitions and routine calculations of bias and variance for estimators. Part (a) requires recall of definitions, part (b) involves simple expectation calculations using linearity, part (c) uses standard variance formulas, and part (d) applies the standard criterion (unbiased + minimum variance). No novel insight or complex problem-solving is required—it's a textbook exercise testing understanding of basic estimator properties. |
| Spec | 5.05b Unbiased estimates: of population mean and variance |
\begin{enumerate}[label=(\alph*)]
\item Explain briefly what you understand by
\begin{enumerate}[label=(\roman*)]
\item an unbiased estimator,
\item a consistent estimator.
\end{enumerate}
\end{enumerate}
of an unknown population parameter $\theta$ [3]
From a binomial population, in which the proportion of successes is $p$, 3 samples of size $n$ are taken. The number of successes $X_1, X_2$, and $X_3$ are recorded and used to estimate $p$.
\begin{enumerate}[label=(\alph*)]
\item[(b)] Determine the bias, if any, of each of the following estimators of $p$.
$\hat{p}_1 = \frac{X_1 + X_2 + X_3}{3n}$,
$\hat{p}_2 = \frac{X_1 + 3X_2 + X_3}{6n}$,
$\hat{p}_3 = \frac{2X_1 + 3X_2 + X_3}{6n}$ [4]
\item Find the variance of each of these estimators. [4]
\item State, giving a reason, which of the three estimators for $p$ is
\begin{enumerate}[label=(\roman*)]
\item the best estimator,
\item the worst estimator. [4]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Edexcel S4 Q5 [15]}}