| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics A AS (Further Statistics A AS) |
| Year | 2021 |
| Session | November |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Poisson distribution |
| Type | Goodness-of-fit test for Poisson |
| Difficulty | Standard +0.3 This is a standard goodness-of-fit test for a Poisson distribution with straightforward calculations: estimating the mean, computing expected frequencies using Poisson probabilities, calculating chi-squared contributions, and comparing to critical values. All steps are routine applications of formulas taught in Further Statistics, requiring no novel insight or complex problem-solving. |
| 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 modelling5.02m Poisson: mean = variance = lambda5.06b Fit prescribed distribution: chi-squared test |
| Number of butterflies | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | \(\geqslant 8\) |
| Frequency | 6 | 9 | 18 | 26 | 13 | 16 | 9 | 3 | 0 |
| Number of butterflies | Frequency | Probability | Expected frequency | Chi-squared contribution |
| 0 | 6 | 0.0369 | 3.6883 | 1.4489 |
| 1 | 9 | 0.1217 | 12.1714 | 0.8264 |
| 2 | 18 | 0.2160 | ||
| 3 | 26 | 0.6916 | ||
| 4 | 13 | 0.1823 | 18.2252 | 1.4981 |
| 5 | 16 | 0.1203 | 12.0286 | |
| 6 | 9 | 0.0662 | 6.6158 | 0.8593 |
| \(\geqslant 7\) | 3 | 0.0510 | 5.0966 | 0.8625 |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (a) | (i) |
| [1] | 1.1 | AG |
| 7 | (a) | (ii) |
| Answer | Marks |
|---|---|
| Because the variance is reasonably close to the mean. | B1* |
| Answer | Marks | Guidance |
|---|---|---|
| [2] | 1.1 | |
| 1.2 | Allow 3.05 (denominator n) | |
| 7 | (b) | DR |
| Answer | Marks |
|---|---|
| = 1.3112 | M1 |
| Answer | Marks |
|---|---|
| [4] | 3.4 |
| Answer | Marks |
|---|---|
| 1.1 | For method using Poisson distribution for either probability |
| Answer | Marks | Guidance |
|---|---|---|
| 7 | (c) | H : Poisson model is a good fit |
| Answer | Marks |
|---|---|
| Poisson model is not a good fit | B1 |
| Answer | Marks |
|---|---|
| [6] | 3.3 |
| Answer | Marks |
|---|---|
| 2.2b | For both hypotheses |
Question 7:
7 | (a) | (i) | 0×6+1×9+⋯.+7×3 330 | B1
[1] | 1.1 | AG
7 | (a) | (ii) | 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = =
Sample variance = 3.08
100 100
Because the variance is reasonably close to the mean. | B1*
depE1
[2] | 1.1
1.2 | Allow 3.05 (denominator n)
7 | (b) | DR
For 2, prob = 0.2008.. so exp freq = 20.0829
For 3, prob = 0.2209.. so exp freq = 22.0912
(16−12.0286)2
For 5, Contribution=
12.0286
= 1.3112 | M1
A1
M1
A1
[4] | 3.4
2.2a
1.1a
1.1 | For method using Poisson distribution for either probability
or expected frequency.
Allow second value to be found by subtraction
SC1 for 1.3112 without M1 earned
7 | (c) | H : Poisson model is a good fit
0
H : Poisson model is not a good fit
1
Re2fer to χ2 :
𝑋𝑋 = 7.7164
critical value at 5% level = 12.59
7.714 < 12.59
There is insufficient evidence to suggest that the
Poisson model is not a good fit | B1
B1
M1
A1
M1
A1
[6] | 3.3
1.1
3.4
1.1
1.1
2.2b | For both hypotheses
No further marks from here if degrees of freedom incorrect.
SC for candidates who merge the first two rows of the table
award B1 for X2 = 5.4853.. M1 for 5 degrees of freedom A1 for
cv = 11.07
χ2(7.714)=0.740<0.95,
Or: which can lead to M1A1
6
M1 for appropriate comparison leading to a conclusion
For conclusion
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recorded or monitored7 A biologist is investigating migrating butterflies. Fig. 7.1 shows the numbers of migrating butterflies passing her location in 100 randomly chosen one-minute periods.
\begin{table}[h]
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | c | c | }
\hline
Number of butterflies & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & $\geqslant 8$ \\
\hline
Frequency & 6 & 9 & 18 & 26 & 13 & 16 & 9 & 3 & 0 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 7.1}
\end{center}
\end{table}
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Use the data to show that a suitable estimate for the mean number of butterflies passing her location per minute is 3.3.
\item Explain how the value of the variance estimate calculated from the sample supports the suggestion that a Poisson distribution may be a suitable model for these data.
The biologist decides to carry out a test to investigate whether a Poisson distribution may be a suitable model for these data.
\end{enumerate}\item In this question you must show detailed reasoning.
Complete the copy of Fig. 7.2 of expected frequencies and contributions for a chi-squared test in the Printed Answer Booklet.
\begin{table}[h]
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
Number of butterflies & Frequency & Probability & Expected frequency & Chi-squared contribution \\
\hline
0 & 6 & 0.0369 & 3.6883 & 1.4489 \\
\hline
1 & 9 & 0.1217 & 12.1714 & 0.8264 \\
\hline
2 & 18 & & & 0.2160 \\
\hline
3 & 26 & & & 0.6916 \\
\hline
4 & 13 & 0.1823 & 18.2252 & 1.4981 \\
\hline
5 & 16 & 0.1203 & 12.0286 & \\
\hline
6 & 9 & 0.0662 & 6.6158 & 0.8593 \\
\hline
$\geqslant 7$ & 3 & 0.0510 & 5.0966 & 0.8625 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 7.2}
\end{center}
\end{table}
\item Complete the chi-squared test at the $5 \%$ significance level.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics A AS 2021 Q7 [13]}}