| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2015 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Goodness-of-fit test for Poisson |
| Difficulty | Standard +0.8 This is a standard chi-squared goodness-of-fit test for a Poisson distribution requiring multiple steps: calculating the sample mean to estimate λ, computing expected frequencies, pooling cells appropriately, calculating the test statistic, determining degrees of freedom (accounting for estimated parameter), and comparing to critical value. While methodical, it's a well-defined procedure covered in Further Maths Statistics with no novel insight required, placing it moderately above average difficulty. |
| Spec | 5.06b Fit prescribed distribution: chi-squared test |
| Number of goals | 0 | 1 | 2 | 3 | 4 | 5 | 6 or more |
| Frequency | 12 | 16 | 31 | 25 | 13 | 3 | 0 |
| Answer | Marks |
|---|---|
| 8 | Find mean of sample data [for use in Poisson distn.]: |
| Answer | Marks | Guidance |
|---|---|---|
| B1 B1 M1 A1 M0 M1 B1 M1 (max 7) | 10 | 10 |
Question 8:
8 | Find mean of sample data [for use in Poisson distn.]:
λ = 220/100 = 2⋅2 B1
State (at least) null hypothesis (AEF): H 0: Poisson distn. fits data
or λ = 2⋅2 B1
Find expected values 100λr e -λ /r! (to 1 d.p.): 11⋅080 24⋅377 26⋅814 19⋅664
(ignore incorrect final value here for M1) 10⋅8151 4⋅759 2⋅491 M1 A1
Combine last two cells so that exp. value ⩾ 5: Oi : 3
Ei : 7⋅25 M1*
Calculate value of χ2 (to 2 d.p.; A1 dep M1*): χ2 = 0⋅076 + 2⋅879 + 0⋅653 + 1⋅448
+ 0⋅441 + 2⋅491
(allow 7⋅95 if 1 d.p. exp.values used) = 7⋅99 M1 A1
State or use consistent tabular value (to 3 s.f.): 5 cells: χ 3 ,0.95 2 = 7⋅815
6 cells: χ 4, 0.95 2 = 9⋅488 (correct)
7 cells: χ 5, 0.95 2 = 11⋅07 B1
State or imply valid method for conclusion e.g.: Accept H0 if χ2 < tabular value M1
Conclusion (AEF, requires both values correct): Distn fits or λ = 2⋅2 A1
Not combining cells [so χ2 = 8⋅64] can earn
B1 B1 M1 A1 M0 M1 B1 M1 (max 7) | 10 | 10The number of goals scored by a certain football team was recorded for each of 100 matches, and the results are summarised in the following table.
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|c|}
\hline
Number of goals & 0 & 1 & 2 & 3 & 4 & 5 & 6 or more \\
\hline
Frequency & 12 & 16 & 31 & 25 & 13 & 3 & 0 \\
\hline
\end{tabular}
\end{center}
Fit a Poisson distribution to the data, and test its goodness of fit at the 5% significance level. [10]
\hfill \mbox{\textit{CAIE FP2 2015 Q8 [10]}}