| Exam Board | OCR |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2016 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared goodness of fit |
| Type | Chi-squared goodness of fit: Other continuous |
| Difficulty | Standard +0.8 This question combines finding a pdf constant, calculating E(X) with a piecewise function, computing expected frequencies from probability integrals, and performing a chi-squared goodness of fit test. While each component is standard S3 material, the piecewise nature adds computational complexity, and students must correctly integrate different functions over different intervals. The multi-step nature and integration of several techniques makes this moderately challenging for A-level Further Maths. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03c Calculate mean/variance: by integration5.06c Fit other distributions: discrete and continuous |
| Time | \(0 \leqslant x < 0.5\) | \(0.5 \leqslant x < 1\) | \(1 \leqslant x < 1.5\) | \(1.5 \leqslant x \leqslant 2\) |
| Observed frequency | 8 | 92 | 279 | 613 |
| Expected frequency | 6 | 90 |
7 A continuous random variable $X$ has probability density function
$$f ( x ) = \begin{cases} a x ^ { 3 } & 0 \leqslant x \leqslant 1 \\ a x ^ { 2 } & 1 < x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$
where $a$ is a constant.\\
(i) Show that $a = \frac { 12 } { 31 }$.\\
(ii) Find $\mathrm { E } ( X )$.
It is thought that the time taken by a student to complete a task can be well modelled by $X$. The times taken by 992 randomly chosen students are summarised in the table, together with some of the expected frequencies.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | }
\hline
Time & $0 \leqslant x < 0.5$ & $0.5 \leqslant x < 1$ & $1 \leqslant x < 1.5$ & $1.5 \leqslant x \leqslant 2$ \\
\hline
Observed frequency & 8 & 92 & 279 & 613 \\
\hline
Expected frequency & 6 & 90 & & \\
\hline
\end{tabular}
\end{center}
(iii) Find the other expected frequencies and test, at the $5 \%$ level of significance, whether the data can be well modelled by $X$.
\hfill \mbox{\textit{OCR S3 2016 Q7 [12]}}