| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2013 |
| Session | June |
| Marks | 9 |
| 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 requires integration of the given pdf to find probabilities, calculation of expected frequencies, chi-squared test execution, and comparison with critical values. While the integration is straightforward (power rule), the multi-step process involving both calculus and hypothesis testing, combined with the Further Maths context and continuous distribution (less routine than discrete), places it moderately above average difficulty. |
| Spec | 5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf5.06b Fit prescribed distribution: chi-squared test |
| Interval | \(2 \leqslant x < 3\) | \(3 \leqslant x < 4\) | \(4 \leqslant x < 5\) | \(5 \leqslant x < 6\) |
| Observed frequency | 36 | 29 | 9 | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(E_2 = 80\int_1^2 3x^{-2}\,dx = 80\big[-3x^{-1}\big]_1^2 = 40\) A.G. | M1 A1 | Find 1st expected frequency \(E_2\) |
| \(E_3 = 20\), \(E_4 = 12\), \(E_5 = 8\) | M1 A1 | Find other expected frequencies |
| \(H_0\): \(f(x)\) fits data (A.E.F.) | B1 | State null hypothesis |
| \(\chi^2 = 0.4 + 4.05 + 0.75 + 0.5 = 5.7\) | M1 A1 | Calculate \(\chi^2\) (to 2 d.p.) |
| \(\chi^2_{3,\,0.9}{}^2 = 6.25[1]\) | *B1 | State or use consistent tabular value |
| \(\chi^2 < 6.25\) so \(f(x)\) does fit | B1\(\sqrt{}\) | Conclusion (A.E.F., \(\sqrt{}\) on \(\chi^2\), dep *B1) |
## Question 7:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $E_2 = 80\int_1^2 3x^{-2}\,dx = 80\big[-3x^{-1}\big]_1^2 = 40$ **A.G.** | M1 A1 | Find 1st expected frequency $E_2$ |
| $E_3 = 20$, $E_4 = 12$, $E_5 = 8$ | M1 A1 | Find other expected frequencies |
| $H_0$: $f(x)$ fits data (A.E.F.) | B1 | State null hypothesis |
| $\chi^2 = 0.4 + 4.05 + 0.75 + 0.5 = 5.7$ | M1 A1 | Calculate $\chi^2$ (to 2 d.p.) |
| $\chi^2_{3,\,0.9}{}^2 = 6.25[1]$ | *B1 | State or use consistent tabular value |
| $\chi^2 < 6.25$ so $f(x)$ does fit | B1$\sqrt{}$ | Conclusion (A.E.F., $\sqrt{}$ on $\chi^2$, dep *B1) |
**Total: 9 marks**
---7 A random sample of 80 observations of the continuous random variable $X$ was taken and the values are summarised in the following table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | }
\hline
Interval & $2 \leqslant x < 3$ & $3 \leqslant x < 4$ & $4 \leqslant x < 5$ & $5 \leqslant x < 6$ \\
\hline
Observed frequency & 36 & 29 & 9 & 6 \\
\hline
\end{tabular}
\end{center}
It is required to test the goodness of fit of the distribution having probability density function f given by
$$f ( x ) = \begin{cases} \frac { 3 } { x ^ { 2 } } & 2 \leqslant x < 6 \\ 0 & \text { otherwise. } \end{cases}$$
Show that the expected frequency for the interval $2 \leqslant x < 3$ is 40 and calculate the remaining expected frequencies.
Carry out a goodness of fit test, at the $10 \%$ significance level.
\hfill \mbox{\textit{CAIE FP2 2013 Q7 [9]}}