OCR MEI Further Statistics Minor 2022 June — Question 3 15 marks

Exam BoardOCR MEI
ModuleFurther Statistics Minor (Further Statistics Minor)
Year2022
SessionJune
Marks15
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicPoisson distribution
TypeGoodness-of-fit test for Poisson
DifficultyStandard +0.8 This is a standard chi-squared goodness-of-fit test for a Poisson distribution, requiring calculation of sample mean, Poisson probabilities, expected frequencies, and chi-squared contributions. While it involves multiple steps and careful arithmetic with the Poisson formula, it follows a well-established procedure taught in Further Statistics. The conceptual demand (understanding why categories are combined, degrees of freedom) is moderate but routine for this specification.
Spec5.02a Discrete probability distributions: general5.02i 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 test5.06c Fit other distributions: discrete and continuous
3 Jane wonders whether the number of wasps entering a wasp's nest per 5 second interval can be modelled by a Poisson distribution with mean \(\mu\). She counts the number of wasps entering the nest over 60 randomly selected 5 -second intervals. The results are shown in Fig. 3.1. \begin{table}[h]
Number of wasps0123456789\(\geqslant 10\)
Frequency025512101011140
\captionsetup{labelformat=empty} \caption{Fig. 3.1}
\end{table}
  1. Show that a suitable estimate for the value of \(\mu\) is 5.1. Fig. 3.2 shows part of a screenshot for a \(\chi ^ { 2 }\) test to assess the goodness of fit of a Poisson model. The sample mean has been used as an estimate for the population mean. Some of the values in the spreadsheet have been deliberately omitted. \begin{table}[h]
    ABCDE
    \includegraphics[max width=\textwidth, alt={}]{e8624e9b-5143-49d2-9683-cc3a1082694e-4_132_40_1069_273}Number of waspsObserved frequencyPoisson probabilityExpected frequencyChi-squared contribution
    2\(\leqslant 2\)70.11656.98870.0000
    3358.08741.1786
    44120.2765
    55100.0255
    66100.14908.94000.1257
    77110.10866.51343.0904
    8\(\geqslant 8\)50.14408.6414
    9
    \captionsetup{labelformat=empty} \caption{Fig. 3.2}
    \end{table}
  2. Determine the missing values in each of the following cells, giving your answers correct to 4 decimal places.
    Carry out the hypothesis test at the 5\% significance level.
  3. Jane also carries out a \(\chi ^ { 2 }\) test for the number of wasps leaving another nest. As part of her calculations, she finds that the probability of no wasps leaving the nest in a 5 -second period is 0.0053 . She finds that a Poisson distribution is also an appropriate model in this case. Find a suitable estimate for the value of the mean number of wasps leaving the nest per 5-second period.
Question 3:
AnswerMarks Guidance
3(a) 1×2+2×5+3×5+4×12+5×10+6×10+7×11+8×1+9×4
60
AnswerMarks
= 5.1M1
A1
AnswerMarks
[2]1.1
1.1May see extra 0 terms in numerator
Accept e.g. 2+10+⋯+36
AnswerMarks
Given answer306
only scores M0
60
Needs to be correctly
obtained
AnswerMarks Guidance
3(b) C3 = 0.1348
D5 = 10.5177
(5−8.6414)2
E8 =
8.6414
AnswerMarks
= 1.5344B1
B1
M1
A1
AnswerMarks
[4]3.4
2.2a
1.1a
AnswerMarks Guidance
1.14 decimal places required for all values
3(c) Because if they were not then (some of) the expected
frequencies would be too low (<5) (and so the test would
AnswerMarks Guidance
not be valid)E1
[1]2.4 To ensure the expected frequencies
large enough
AnswerMarks Guidance
3(d) H : Poisson model is a good fit
0
H : Poisson model is not a good fit
1
X2 = 6.23
Refer to X2
5
Critical value at 5% level = 11.07
6.23 < 11.07 (Accept H )
0
There is insufficient evidence to suggest that the
Poisson model is not a good fit for (the number of)
AnswerMarks
wasps (entering the nests).B1
B1FT
M1
A1
M1
A1
AnswerMarks
[6]1.2
1.1
3.4
1.1
AnswerMarks
2.2bReference to ‘mean 5.1’ in hypotheses
Scores B0
FT their value of E8
or 𝜒2(6.231) = 0.7156
5
0.7156 < 0.95
Correct test and critical values required
Conclusion must follow correct
hypotheses, not be too assertive and
AnswerMarks
refer to context.Allow omission of
context at this stage
Comparing their test
and critical values
leading to a
conclusion.
AnswerMarks Guidance
3(e) 𝜆0
e−𝜆( ) = 0.0053
0!
AnswerMarks
λ = 5.2(4004…)M1
A1
AnswerMarks
[2]3.1a
1.1
Question 3:
3 | (a) | 1×2+2×5+3×5+4×12+5×10+6×10+7×11+8×1+9×4
60
= 5.1 | M1
A1
[2] | 1.1
1.1 | May see extra 0 terms in numerator
Accept e.g. 2+10+⋯+36
Given answer | 306
only scores M0
60
Needs to be correctly
obtained
3 | (b) | C3 = 0.1348
D5 = 10.5177
(5−8.6414)2
E8 =
8.6414
= 1.5344 | B1
B1
M1
A1
[4] | 3.4
2.2a
1.1a
1.1 | 4 decimal places required for all values
3 | (c) | Because if they were not then (some of) the expected
frequencies would be too low (<5) (and so the test would
not be valid) | E1
[1] | 2.4 | To ensure the expected frequencies
large enough
3 | (d) | H : Poisson model is a good fit
0
H : Poisson model is not a good fit
1
X2 = 6.23
Refer to X2
5
Critical value at 5% level = 11.07
6.23 < 11.07 (Accept H )
0
There is insufficient evidence to suggest that the
Poisson model is not a good fit for (the number of)
wasps (entering the nests). | B1
B1FT
M1
A1
M1
A1
[6] | 1.2
1.1
3.4
1.1
2.2b | Reference to ‘mean 5.1’ in hypotheses
Scores B0
FT their value of E8
or 𝜒2(6.231) = 0.7156
5
0.7156 < 0.95
Correct test and critical values required
Conclusion must follow correct
hypotheses, not be too assertive and
refer to context. | Allow omission of
context at this stage
Comparing their test
and critical values
leading to a
conclusion.
3 | (e) | 𝜆0
e−𝜆( ) = 0.0053
0!
λ = 5.2(4004…) | M1
A1
[2] | 3.1a
1.1
3 Jane wonders whether the number of wasps entering a wasp's nest per 5 second interval can be modelled by a Poisson distribution with mean $\mu$. She counts the number of wasps entering the nest over 60 randomly selected 5 -second intervals. The results are shown in Fig. 3.1.

\begin{table}[h]
\begin{center}
\begin{tabular}{ | l | l | l | l | l | c | c | c | c | l | l | c | }
\hline
Number of wasps & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & $\geqslant 10$ \\
\hline
Frequency & 0 & 2 & 5 & 5 & 12 & 10 & 10 & 11 & 1 & 4 & 0 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 3.1}
\end{center}
\end{table}
\begin{enumerate}[label=(\alph*)]
\item Show that a suitable estimate for the value of $\mu$ is 5.1.

Fig. 3.2 shows part of a screenshot for a $\chi ^ { 2 }$ test to assess the goodness of fit of a Poisson model. The sample mean has been used as an estimate for the population mean. Some of the values in the spreadsheet have been deliberately omitted.

\begin{table}[h]
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
 & A & B & C & D & E \\
\hline
\includegraphics[max width=\textwidth, alt={}]{e8624e9b-5143-49d2-9683-cc3a1082694e-4_132_40_1069_273}
 & Number of wasps & Observed frequency & Poisson probability & Expected frequency & Chi-squared contribution \\
\hline
2 & $\leqslant 2$ & 7 & 0.1165 & 6.9887 & 0.0000 \\
\hline
3 & 3 & 5 &  & 8.0874 & 1.1786 \\
\hline
4 & 4 & 12 &  &  & 0.2765 \\
\hline
5 & 5 & 10 &  &  & 0.0255 \\
\hline
6 & 6 & 10 & 0.1490 & 8.9400 & 0.1257 \\
\hline
7 & 7 & 11 & 0.1086 & 6.5134 & 3.0904 \\
\hline
8 & $\geqslant 8$ & 5 & 0.1440 & 8.6414 &  \\
\hline
9 &  &  &  &  &  \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 3.2}
\end{center}
\end{table}
\item Determine the missing values in each of the following cells, giving your answers correct to 4 decimal places.

\begin{itemize}
  \item C3
  \item D5
  \item E8
\item Explain why some of the frequencies have been combined into the categories $\leqslant 2$ and $\geqslant 8$.
\item In this question you must show detailed reasoning.
\end{itemize}

Carry out the hypothesis test at the 5\% significance level.
\item Jane also carries out a $\chi ^ { 2 }$ test for the number of wasps leaving another nest. As part of her calculations, she finds that the probability of no wasps leaving the nest in a 5 -second period is 0.0053 . She finds that a Poisson distribution is also an appropriate model in this case.

Find a suitable estimate for the value of the mean number of wasps leaving the nest per 5-second period.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Statistics Minor 2022 Q3 [15]}}
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