| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2020 |
| Session | October |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Uniform Distribution |
| Type | Find parameter from variance or other constraint |
| Difficulty | Standard +0.3 This is a straightforward application of bias definition requiring students to show E(2X̄) ≠ α using the known mean (α+1)/2 of a discrete uniform distribution, followed by a simple calculation of sample mean and estimation. The concepts are standard S3 material with minimal problem-solving required beyond applying formulas. |
| Spec | 5.05b Unbiased estimates: of population mean and variance |
\begin{enumerate}
\item The random variable $X$ has the discrete uniform distribution
\end{enumerate}
$$\mathrm { P } ( X = x ) = \frac { 1 } { \alpha } \quad \text { for } x = 1,2 , \ldots , \alpha$$
The mean of a random sample of size $n$, taken from this distribution, is denoted by $\bar { X }$\\
(a) Show that $2 \bar { X }$ is a biased estimator of $\alpha$
A random sample of 6 observations of $X$ is taken and the results are given below.
$$\begin{array} { l l l l l l }
8 & 7 & 3 & 7 & 2 & 9
\end{array}$$
(b) Use the sample mean to estimate the value of $\alpha$\\
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\hfill \mbox{\textit{Edexcel S3 2020 Q1 [4]}}