| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared goodness of fit |
| Type | Assess model suitability before testing |
| Difficulty | Standard +0.3 This is a standard chi-squared goodness of fit question with routine calculations: finding sample mean/variance, computing Poisson probabilities, and performing a hypothesis test. The preliminary model assessment (comparing mean and variance) is straightforward, and the test procedure follows a standard template. While it requires multiple steps and careful arithmetic, it demands no novel insight beyond applying learned procedures. |
| Spec | 5.02i Poisson distribution: random events model5.02j Poisson formula: P(X=x) = e^(-lambda)*lambda^x/x!5.02k Calculate Poisson probabilities5.02l Poisson conditions: for modelling5.06b Fit prescribed distribution: chi-squared test |
| Number of letters | 0 | 1 | 2 | 3 | 4 | 5 | \(\geqslant 6\) |
| Number of days | 57 | 60 | 53 | 25 | 4 | 1 | 0 |
| Number of letters | 0 | 1 | 2 | 3 | 4 | 5 | \(\geqslant 6\) |
| Expected number of days | 53.964 | 70.693 | \(p\) | \(q\) | 6.622 | 1.735 | 0.463 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Resolve vertically at equilibrium: \(\lambda d/a = mg\) \([\lambda/a = mg/d]\) | B1 | |
| Newton's Law at general point: \(m\,d^2x/dt^2 = mg - \lambda(d+x)/a\) | ||
| \([or\ -mg + \lambda(d-x)/a]\) | M1 A1 | |
| Simplify: \(d^2x/dt^2 = -(\lambda/ma)\,x\) *or* \(-(g/d)\,x\) | A1 | |
| Stating without derivation (max 3/5) | (B1) | |
| Find period \(T\) using SHM \(\omega = \sqrt{g/d}\): \(T = [2\pi\sqrt{(ma/\lambda)}] = 2\pi\sqrt{(d/g)}\) A.G. | B1 | 5 marks total |
| Use SHM formula with amplitude \(2d\): \(x = 2d\cos(\omega t)\) \([or\ \sin]\) | M1 | |
| Find time \(t_1\) to string becoming slack: \(t_1 = (1/\omega)\cos^{-1}(-1/2)\) | ||
| *or* \(T/4 + (1/\omega)\sin^{-1}(1/2)\) | M1 A1 | |
| A.G. \(t_1 = (1/\omega)\,2\pi/3 = (2\pi/3)\sqrt{(d/g)}\) | A1 | |
| Find speed \(v\) when string slack: \(v = \omega\sqrt{(4d^2 - d^2)} = \omega d\sqrt{3}\) *or* \(\sqrt{(3dg)}\) | M1 A1 | |
| Find further time \(t_2\) to instantaneous rest: \(t_2 = v/g\) | B1 | |
| A.G. \(t_2 = \sqrt{(3dg)}/g = \sqrt{3}\sqrt{(d/g)}\) | M1 A1 | 9 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Find mean and variance: \(262/200 = 1.31\) *and* \((586 - 262^2/200)/200 = 1.21\) | M1 A1 | |
| Values close, so distribution appropriate | B1 | 3 marks total; AEF, needs values approx correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| A.G.: \(p = 200(1.31^2/2)e^{-1.31} = 46.304\) | B1 | |
| \(q = 200(1.31^3/6)e^{-1.31} = 20.2\) | B1 | 2 marks total; can use \(\sum E_i = 200\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(H_0\): Poisson fits data | B1 | State at least null hypothesis |
| Combine last 3 cells since exp. value \(< 5\): \(O: \ldots\ 5\) | ||
| \(E: \ldots\ 8.82\) | *M1 A1 | |
| \(\chi^2 = 5.54\) | M1 A1 | Calculate \(\chi^2\) to 2 dp; A1 dep *M1 |
| \(\chi^2_{3,\,0.9} = 6.251\) | M1 A1 | Compare consistent tabular value to 2 dp |
| \([\chi^2_{4,\,0.9} = 7.779,\ \chi^2_{5,\,0.9} = 9.236]\) | A1 dep *M1 | |
| Accept \(H_0\) if \(\chi^2 <\) tabular value | M1 | Valid method for conclusion |
| \(5.54 < 6.25\) so Poisson does fit | A1 | 9 marks total; A.E.F., needs correct values |
# Question 10:
## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| Resolve vertically at equilibrium: $\lambda d/a = mg$ $[\lambda/a = mg/d]$ | B1 | |
| Newton's Law at general point: $m\,d^2x/dt^2 = mg - \lambda(d+x)/a$ | | |
| $[or\ -mg + \lambda(d-x)/a]$ | M1 A1 | |
| Simplify: $d^2x/dt^2 = -(\lambda/ma)\,x$ *or* $-(g/d)\,x$ | A1 | |
| Stating without derivation (max 3/5) | (B1) | |
| Find period $T$ using SHM $\omega = \sqrt{g/d}$: $T = [2\pi\sqrt{(ma/\lambda)}] = 2\pi\sqrt{(d/g)}$ **A.G.** | B1 | 5 marks total |
| Use SHM formula with amplitude $2d$: $x = 2d\cos(\omega t)$ $[or\ \sin]$ | M1 | |
| Find time $t_1$ to string becoming slack: $t_1 = (1/\omega)\cos^{-1}(-1/2)$ | | |
| *or* $T/4 + (1/\omega)\sin^{-1}(1/2)$ | M1 A1 | |
| **A.G.** $t_1 = (1/\omega)\,2\pi/3 = (2\pi/3)\sqrt{(d/g)}$ | A1 | |
| Find speed $v$ when string slack: $v = \omega\sqrt{(4d^2 - d^2)} = \omega d\sqrt{3}$ *or* $\sqrt{(3dg)}$ | M1 A1 | |
| Find further time $t_2$ to instantaneous rest: $t_2 = v/g$ | B1 | |
| **A.G.** $t_2 = \sqrt{(3dg)}/g = \sqrt{3}\sqrt{(d/g)}$ | M1 A1 | 9 marks total |
## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| Find mean and variance: $262/200 = 1.31$ *and* $(586 - 262^2/200)/200 = 1.21$ | M1 A1 | |
| Values close, so distribution appropriate | B1 | 3 marks total; AEF, needs values approx correct |
## Part (b)(i)
| Answer | Mark | Guidance |
|--------|------|----------|
| **A.G.:** $p = 200(1.31^2/2)e^{-1.31} = 46.304$ | B1 | |
| $q = 200(1.31^3/6)e^{-1.31} = 20.2$ | B1 | 2 marks total; can use $\sum E_i = 200$ |
## Part (b)(ii)
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0$: Poisson fits data | B1 | State at least null hypothesis |
| Combine last 3 cells since exp. value $< 5$: $O: \ldots\ 5$ | | |
| $E: \ldots\ 8.82$ | *M1 A1 | |
| $\chi^2 = 5.54$ | M1 A1 | Calculate $\chi^2$ to 2 dp; A1 dep *M1 |
| $\chi^2_{3,\,0.9} = 6.251$ | M1 A1 | Compare consistent tabular value to 2 dp |
| $[\chi^2_{4,\,0.9} = 7.779,\ \chi^2_{5,\,0.9} = 9.236]$ | | A1 dep *M1 |
| Accept $H_0$ if $\chi^2 <$ tabular value | M1 | Valid method for conclusion |
| $5.54 < 6.25$ so Poisson does fit | A1 | 9 marks total; A.E.F., needs correct values |A family was asked to record the number of letters delivered to their house on each of 200 randomly chosen weekdays. The results are summarised in the following table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | }
\hline
Number of letters & 0 & 1 & 2 & 3 & 4 & 5 & $\geqslant 6$ \\
\hline
Number of days & 57 & 60 & 53 & 25 & 4 & 1 & 0 \\
\hline
\end{tabular}
\end{center}
It is suggested that the number of letters delivered each weekday has a Poisson distribution. By finding the mean and variance for this sample, comment on the appropriateness of this suggestion.
The following table includes some of the expected values, correct to 3 decimal places, using a Poisson distribution with mean equal to the sample mean for the above data.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | }
\hline
Number of letters & 0 & 1 & 2 & 3 & 4 & 5 & $\geqslant 6$ \\
\hline
Expected number of days & 53.964 & 70.693 & $p$ & $q$ & 6.622 & 1.735 & 0.463 \\
\hline
\end{tabular}
\end{center}
(i) Show that $p = 46.304$, correct to 3 decimal places, and find $q$.\\
(ii) Carry out a goodness of fit test at the $10 \%$ significance level.
\hfill \mbox{\textit{CAIE FP2 2011 Q10 OR}}