| Exam Board | OCR MEI |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2006 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared goodness of fit |
| Type | Chi-squared goodness of fit: Other continuous |
| Difficulty | Standard +0.3 This is a straightforward chi-squared goodness of fit test with most work done for you. Part (i) requires standard calculus (mean via integration, mode via differentiation). Part (ii) is routine integration with verification provided. Part (iii) is a textbook chi-squared test where expected frequencies are directly calculable from the given CDF values—no degrees of freedom ambiguity or complex interpretation needed. Slightly easier than average due to the scaffolding and verification steps. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.03e Find cdf: by integration5.06b Fit prescribed distribution: chi-squared test |
| Load \(x\) | \(0 \leqslant x \leqslant \frac { 1 } { 4 }\) | \(\frac { 1 } { 4 } < x \leqslant \frac { 1 } { 2 }\) | \(\frac { 1 } { 2 } < x \leqslant \frac { 3 } { 4 }\) | \(\frac { 3 } { 4 } < x \leqslant 1\) |
| Frequency | 126 | 209 | 131 | 46 |
1 Design engineers are simulating the load on a particular part of a complex structure. They intend that the simulated load, measured in a convenient unit, should be given by the random variable $X$ having probability density function
$$f ( x ) = 12 x ^ { 3 } - 24 x ^ { 2 } + 12 x , \quad 0 \leqslant x \leqslant 1 .$$
(i) Find the mean and the mode of $X$.\\
(ii) Find the cumulative distribution function $\mathrm { F } ( x )$ of $X$.
$$\text { Verify that } \mathrm { F } \left( \frac { 1 } { 4 } \right) = \frac { 67 } { 256 } , \mathrm {~F} \left( \frac { 1 } { 2 } \right) = \frac { 11 } { 16 } \text { and } \mathrm { F } \left( \frac { 3 } { 4 } \right) = \frac { 243 } { 256 } .$$
The engineers suspect that the process for generating simulated loads might not be working as intended. To investigate this, they generate a random sample of 512 loads. These are recorded in a frequency distribution as follows.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | }
\hline
Load $x$ & $0 \leqslant x \leqslant \frac { 1 } { 4 }$ & $\frac { 1 } { 4 } < x \leqslant \frac { 1 } { 2 }$ & $\frac { 1 } { 2 } < x \leqslant \frac { 3 } { 4 }$ & $\frac { 3 } { 4 } < x \leqslant 1$ \\
\hline
Frequency & 126 & 209 & 131 & 46 \\
\hline
\end{tabular}
\end{center}
(iii) Use a suitable statistical procedure to assess the goodness of fit of $X$ to these data. Discuss your conclusions briefly.
\hfill \mbox{\textit{OCR MEI S3 2006 Q1 [18]}}