| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2017 |
| Session | Specimen |
| Marks | 15 |
| Topic | Exponential Distribution |
| Type | Calculate mean or variance |
| Difficulty | Standard +0.8 This is a Further Maths statistics question requiring calculation of exponential distribution parameters, verification of expected frequencies using integration/CDF, and a chi-squared goodness of fit test with appropriate pooling and degrees of freedom adjustment. While each component is standard, the combination of continuous distribution theory, integration, and hypothesis testing in a multi-part question with 15 marks elevates it above typical A-level questions. |
| Spec | 5.03b Solve problems: using pdf5.03c Calculate mean/variance: by integration5.06c Fit other distributions: discrete and continuous |
| Range | \(0 \leq x < 1\) | \(1 \leq x < 2\) | \(2 \leq x < 3\) | \(3 \leq x < 4\) | \(x \geq 4\) |
| Observed | 24 | 22 | 10 | 3 | 1 |
| Expected | 33.040 | 14.846 | 6.671 | 2.997 | 2.446 |
A continuous random variable $X$ has probability density function given by
$$f(x) = \begin{cases}
0.8e^{-0.8x} & x \geq 0, \\
0 & x < 0.
\end{cases}$$
\begin{enumerate}[label=(\roman*)]
\item Find the mean and variance of $X$. [4]
\end{enumerate}
The lifetime of a certain organism is thought to have the same distribution as $X$. The lifetimes in days of a random sample of 60 specimens of the organism were found. The observed frequencies, together with the expected frequencies correct to 3 decimal places, are given in the table.
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Range & $0 \leq x < 1$ & $1 \leq x < 2$ & $2 \leq x < 3$ & $3 \leq x < 4$ & $x \geq 4$ \\
\hline
Observed & 24 & 22 & 10 & 3 & 1 \\
\hline
Expected & 33.040 & 14.846 & 6.671 & 2.997 & 2.446 \\
\hline
\end{tabular}
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Show how the expected frequency for $1 \leq x < 2$ is obtained. [4]
\item Carry out a goodness of fit test at the 5\% significance level. [7]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics 2017 Q8 [15]}}