| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics Minor (Further Statistics Minor) |
| Session | Specimen |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared goodness of fit |
| Type | Spreadsheet-based chi-squared test |
| Difficulty | Standard +0.3 This is a standard Further Statistics question on Poisson goodness-of-fit testing with straightforward calculations: variance from frequency table, chi-squared test mechanics, and working backwards from probabilities to find a parameter. All parts follow textbook procedures with no novel problem-solving required, though it's slightly above average difficulty due to being Further Maths content and requiring careful attention to multiple computational steps. |
| Spec | 5.02a Discrete probability distributions: general5.06b Fit prescribed distribution: chi-squared test5.06c Fit other distributions: discrete and continuous5.06d Goodness of fit: chi-squared test |
| Number of recaptures, \(x\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Frequency | 2 | 5 | 5 | 9 | 10 | 4 | 3 | 1 | 0 | 1 | 0 |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (i) | Standard deviation = 1.90546... |
| Sample variance = 3.63 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| [2] | 1.1 | |
| 1.1 | BC | SC1 for square of other |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (ii) | Mean not too far from variance so Poisson may be |
| suitable | E1 | |
| [1] | 2.2b | N |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (iii) | H : Poisson model is a good fit |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | B1 | |
| [1] | 2.5 | E |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (iv) | Cell C3 = 0.1929 |
| Answer | Marks |
|---|---|
| (cid:32)0.2200 | B1 |
| Answer | Marks |
|---|---|
| [4] | M |
| Answer | Marks | Guidance |
|---|---|---|
| 1.1 | 40 times their C3 | |
| 6 | (v) | E |
| Answer | Marks |
|---|---|
| model is not a good fit. | B1 |
| Answer | Marks |
|---|---|
| [5] | 1.1 |
| Answer | Marks |
|---|---|
| 2.2b | Degrees of freedom = 4 soi |
| Answer | Marks |
|---|---|
| Conclusion | last two marks allowed if |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (vi) | e(cid:16)(cid:79)(cid:79) e(cid:16)(cid:79)(cid:79)2 |
| Answer | Marks |
|---|---|
| (cid:79)(cid:32)2.1 | M1 |
| Answer | Marks |
|---|---|
| [3] | 3.1b |
| Answer | Marks |
|---|---|
| 1.1 | e(cid:16)(cid:79)(cid:79)2 |
Question 6:
6 | (i) | Standard deviation = 1.90546...
Sample variance = 3.63 | M1
A1
[2] | 1.1
1.1 | BC | SC1 for square of other
“standard deviation” to
get 3.54
6 | (ii) | Mean not too far from variance so Poisson may be
suitable | E1
[1] | 2.2b | N
Not “Poisson is good fit”
(must be non-assertive).
6 | (iii) | H : Poisson model is a good fit
0
H : Poisson model is not a good fit
1 | B1
[1] | 2.5 | E
Ignore any reference to the
value of the Poisson parameter
6 | (iv) | Cell C3 = 0.1929
Cell D3 = 7.7159
(cid:11)4(cid:16)5.0544(cid:12)2
Cell E6 (cid:32)
5.0544
(cid:32)0.2200 | B1
I
B1FT
C
M1
A1
[4] | M
3.4
2.2a
1.1a
1.1 | 40 times their C3
6 | (v) | E
X 2 = 2.292
P
Refer to X 2
4
Critical value at 10% level = 7.779
2.292 < 7.779 S
Do not reject H
0
There is insufficient evidence to suggest that the Poisson
model is not a good fit. | B1
M1
A1
M1
A1
[5] | 1.1
3.4
1.1
1.1
2.2b | Degrees of freedom = 4 soi
Comparison with critical value
Conclusion | last two marks allowed if
5 degrees of freedom used
6 | (vi) | e(cid:16)(cid:79)(cid:79) e(cid:16)(cid:79)(cid:79)2
(cid:32)0.25716 and (cid:32)0.27002
1 2
(cid:79) 0.27002
(cid:32)
2 0.25716
(cid:79)(cid:32)2.1 | M1
M1
A1
[3] | 3.1b
2.2a
1.1 | e(cid:16)(cid:79)(cid:79)2
OR (cid:32)0.27002 and
2
e(cid:16)(cid:79)(cid:79)3
N(cid:32)0.18901
3!
(cid:79) 0.18901
OR (cid:32)
3 0.27002At a bird feeding station, birds are captured and ringed. If a bird is recaptured, the ring enables it to be identified. The table below shows the number of recaptures, $x$, during a period of a month, for each bird of a particular species in a random sample of $40$ birds.
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Number of recaptures, $x$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
Frequency & 2 & 5 & 5 & 9 & 10 & 4 & 3 & 1 & 0 & 1 & 0 \\
\hline
\end{tabular}
\begin{enumerate}[label=(\roman*)]
\item The sample mean of $x$ is $3.4$. Calculate the sample variance of $x$. [2]
\item Briefly comment on whether the results of part (i) support a suggestion that a Poisson model might be a good fit to the data. [1]
\end{enumerate}
The screenshot below shows part of a spreadsheet for a $\chi^2$ test to assess the goodness of fit of a Poisson model. The sample mean of $3.4$ has been used as an estimate of the Poisson parameter. Some values in the spreadsheet have been deliberately omitted.
\includegraphics{figure_2}
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item State the null and alternative hypotheses for the test. [1]
\item Calculate the missing values in cells
\begin{itemize}
\item C3,
\item D3 and
\item E6. [4]
\end{itemize}
\item Complete the test at the $10\%$ significance level. [5]
\item The screenshot below shows part of a spreadsheet for a $\chi^2$ test for a different species of bird. Find the value of the Poisson parameter used.
\includegraphics{figure_3}
[3]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics Minor Q6 [16]}}