| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics Major (Further Statistics Major) |
| Year | 2023 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Standard 2×2 contingency table |
| Difficulty | Standard +0.3 This is a straightforward chi-squared test of independence with a standard 2×2 contingency table. Part (a) requires routine application of the chi-squared test procedure (calculate expected frequencies, compute test statistic, compare to critical value), which is a standard Further Statistics topic. Part (b) asks for a simple conceptual explanation about bias. While it's a Further Maths topic, the question follows a completely standard template with no novel insight required, making it slightly easier than average overall. |
| Spec | 5.06a Chi-squared: contingency tables |
| Where registered | |||||
| \cline { 3 - 4 } \multicolumn{2}{|c|}{} | Local | Non-local | |||
\multirow{2}{*}{
| Under 1.5 m | 12 | 11 | ||
| \cline { 2 - 4 } | At least 1.5 m | 157 | 70 | ||
| Answer | Marks | Guidance |
|---|---|---|
| 9 | (a) | DR |
| Answer | Marks |
|---|---|
| distance. | B1 |
| Answer | Marks |
|---|---|
| [9] | 3.3 |
| Answer | Marks |
|---|---|
| 3.5a | Allow hypotheses and conclusion in terms of independence, |
| Answer | Marks | Guidance |
|---|---|---|
| 9 | (b) | Because the cyclist will know whether or not the car is |
| locally registered when she notes the passing distance | E1 | |
| [1] | 2.2b | Allow other reasonable answers |
| Answer | Marks | Guidance |
|---|---|---|
| Expected | Local | Non-local |
| Under 1.5m | 15.55 | 7.45 |
| At least 1.5m | 153.45 | 73.55 |
| Contribution | Local | Non-local |
| Under 1.5m | 0.8096 | 1.6893 |
| At least 1.5m | 0.0820 | 0.1712 |
| Contribution | Local | Non-local |
| Under 1.5m | 0.5975 | 1.2467 |
| At least 1.5m | 0.0605 | 0.1263 |
Question 9:
9 | (a) | DR
H : no association between where registered and passing
0
distance
H : some association between where registered and
1
passing distance
Expected Local Non-local
Under 1.5m 15.55 7.45
At least 1.5m 153.45 73.55
Contribution Local Non-local
Under 1.5m 0.8096 1.6893
At least 1.5m 0.0820 0.1712
X2=2.75
Use of 𝜒2
1
Critical value at 5% level = 3.84;
2.75 < 3.84 so not significant (do not reject H )
0
There is insufficient evidence to suggest that there is an
association between where registered and passing
distance. | B1
M1
A1
B1
B1
B1
B1
M1
E1
[9] | 3.3
3.4
1.1
1.1
1.1
2.5
2.2b
2.2b
3.5a | Allow hypotheses and conclusion in terms of independence,
but not relationship or correlation which results in B0 and
final E0 Context needed in at least one of the hypotheses
For at least 1 correct expectation
For all correct
At least one correct value must be seen; this mark cannot be
implied merely by a correct final value of X 2. Allow 1 error.
Allow 2.76 from expected values given to 2dp
Yates’ correction is not expected, but full marks if used
correctly
Contribution Local Non-local
Under 1.5m 0.5975 1.2467
At least 1.5m 0.0605 0.1263
X 2 = 2 .0 3
Can be implied by correct critical value of 3.84 or by 2.706
or 5.024. Must be from chi-squared with 1 deg of freedom
OR p-value = 0.09725 and compare with 5%
FT their sensibly obtained test statistic and cv for M1 but
not for A1
If hypotheses wrong way around do not allow first or last
two marks. Must mention ‘insufficient evidence’ in
conclusion, not just ‘insufficient evidence to reject H ’
0
followed by a completely new sentence.
9 | (b) | Because the cyclist will know whether or not the car is
locally registered when she notes the passing distance | E1
[1] | 2.2b | Allow other reasonable answers
Do not allow ‘She may not be able to estimate accurately’
Expected | Local | Non-local
Under 1.5m | 15.55 | 7.45
At least 1.5m | 153.45 | 73.55
Contribution | Local | Non-local
Under 1.5m | 0.8096 | 1.6893
At least 1.5m | 0.0820 | 0.1712
Contribution | Local | Non-local
Under 1.5m | 0.5975 | 1.2467
At least 1.5m | 0.0605 | 0.12639 A cyclist who lives on an island suspects that car drivers with locally registered number plates allow more space when passing her than those with non-locally registered number plates. She decides to carry out a hypothesis test and so over a period of time selects a random sample of 250 cars which pass her. For each car she estimates whether the car driver allows at least the recommended 1.5 metres when passing her. The table shows the data which she collected.
\begin{center}
\begin{tabular}{ | c | l | c | c | }
\hline
\multicolumn{2}{|c|}{} & \multicolumn{2}{c|}{Where registered} \\
\cline { 3 - 4 }
\multicolumn{2}{|c|}{} & Local & Non-local \\
\hline
\multirow{2}{*}{\begin{tabular}{ c }
Passing \\
distance \\
\end{tabular}} & Under 1.5 m & 12 & 11 \\
\cline { 2 - 4 }
& At least 1.5 m & 157 & 70 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item In this question you must show detailed reasoning.
Carry out the test at the $5 \%$ significance level to examine whether there is any association between where the car is registered and passing distance.
\item A friend of the cyclist suggests that there may be a problem with the data, since the cyclist may have introduced some bias in estimating whether cars were allowing the recommended distance.
Explain how any bias might have arisen.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics Major 2023 Q9 [10]}}