| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics Major (Further Statistics Major) |
| Year | 2024 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Expected frequencies partially provided |
| Difficulty | Standard +0.3 This is a straightforward chi-squared test of independence question requiring standard calculations: computing expected frequencies using row×column totals divided by grand total, then calculating one contribution to the test statistic using (O-E)²/E. These are routine A-level Further Statistics procedures with no conceptual challenges or novel problem-solving required. |
| Spec | 5.06a Chi-squared: contingency tables |
| \multirow{2}{*}{} | Bicycle | ||||
| Road | Gravel | Electric | Total | ||
| \multirow{2}{*}{Maximum speed} | Less than 25 mph | 2 | 21 | 19 | 42 |
| 25 mph or greater | 13 | 47 | 18 | 78 | |
| Total | 15 | 68 | 37 | 120 | |
| \multirow{2}{*}{Expected frequency} | Bicycle | |||
| Road | Gravel | Electric | ||
| \multirow{2}{*}{Maximum speed} | Less than 25 mph | 12.95 | ||
| 25 mph or greater | 24.05 | |||
| \multirow{2}{*}{Contribution to the test statistic} | Bicycle | |||
| Road | Gravel | Electric | ||
| \multirow{2}{*}{Maximum speed} | Less than 25 mph | 2.0119 | 0.3294 | 2.8264 |
| 25 mph or greater | 1.0833 | 0.1774 | ||
| Answer | Marks | Guidance |
|---|---|---|
| 9 | (a) | 42×15 42×68 |
| Answer | Marks |
|---|---|
| 120 120 | B1 |
| Answer | Marks |
|---|---|
| [2] | 3.4 |
| 1.1 | B1 for any one correct |
| Answer | Marks | Guidance |
|---|---|---|
| 9 | (b) | (18−24.05)2 |
| Answer | Marks |
|---|---|
| = 1.5219 | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 1.1 | (𝑂−𝐸)2 |
| Answer | Marks | Guidance |
|---|---|---|
| 9 | (c) | DR |
| Answer | Marks |
|---|---|
| Please annotate page 15 | B1 |
| Answer | Marks |
|---|---|
| [6] | 2.5 |
| Answer | Marks |
|---|---|
| 3.5a | For both. Allow independent/not independent. Do NOT |
| Answer | Marks | Guidance |
|---|---|---|
| 9 | (d) | For the road bike, the contribution of 2.0119 shows that fewer |
| Answer | Marks |
|---|---|
| than expected have max speed ≥ 25 mph. | B1 |
| Answer | Marks |
|---|---|
| [3] | 2.2b |
| Answer | Marks |
|---|---|
| 3.5a | Do not allow comments which suggest that gravel has more |
Question 9:
9 | (a) | 42×15 42×68
Road < 25 = = 5.25 Gravel < 25 = = 23.80
120 120
78×15 78×68
Road ≥ 25 = = 9.75 Gravel ≥ 25 = = 44.20
120 120 | B1
B1
[2] | 3.4
1.1 | B1 for any one correct
B1 for the other 3 (perhaps by subtraction)
9 | (b) | (18−24.05)2
24.05
= 1.5219 | M1
A1
[2] | 1.1
1.1 | (𝑂−𝐸)2
For
𝐸
Must be given to 4d.p.
9 | (c) | DR
H : no association between type of bicycle and (whether the)
0
maximum speed (is 25mph or greater).
H : some association between type of bicycle and (whether
1
the) maximum speed (is 25mph or greater).
Test statistic = 7.9503
Degrees of freedom = 2
Critical value = 5.991
7.9503 > 5.991 so reject H
0
There is sufficient evidence to suggest that there is some
association between type of bicycle and (whether the)
maximum speed (is 25mph or greater).
Please annotate page 15 | B1
B1
B1
B1
M1
A1
[6] | 2.5
1.1
3.3
1.1
2.2b
3.5a | For both. Allow independent/not independent. Do NOT
allow relationship in place of association
BC
Can be implied by correct critical value or p-value
Or p-value = 0.0188 Or 𝛸2(7.9503) = 0.9812
2
For comparison (or 0.0188 < 0.05 or 0.9812 > 0.95 oe)
Must have correct test statistic and critical value
If hypotheses wrong way around allow MAX
B0B1B1B1M0A0
9 | (d) | For the road bike, the contribution of 2.0119 shows that fewer
journeys than expected have max speed < 25 mph whereas the
contribution of 1.0833 shows that more journeys than expected
have max speed ≥ 25 mph.
For the gravel bike, the levels are as expected.
For the electric bike, the contribution of 2.8264 shows that
more journeys than expected have max speed < 25 mph
whereas the contribution of 1.5219 shows that fewer journeys
than expected have max speed ≥ 25 mph. | B1
B1
B1
[3] | 2.2b
3.5a
3.5a | Do not allow comments which suggest that gravel has more
over 25 than expected and fewer 25 or less unless mention
that contributions are (very) small so effect is very slight
If B0B0B0 or if more marks gained below then
SC B2 for 5 correct comments without mentioning the
actual values of the contributions or the size of
contributions and no incorrect comments
SC B1 for any 2 correct comments without mentioning
contributions
SCB2 for one correct comment in each case and mentioning
actual values of the contributions or the size of
contributions.9 A cyclist has 3 bicycles, a road bike, a gravel bike and an electric bike. She wishes to know if the bicycle which she is riding makes any difference to whether she reaches a speed of 25 mph or greater on a journey. She selects a random sample of 120 journeys and notes the bicycle and whether or not her maximum speed was 25 mph or greater. She decides to carry out a chisquared test to investigate whether there is any association between bicycle type and whether her maximum speed is 25 mph or greater. Tables 9.1 and 9.2 show the data and some of the expected frequencies for the test.
\begin{table}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Table 9.1}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{}} & \multicolumn{3}{|c|}{Bicycle} & \\
\hline
& & Road & Gravel & Electric & Total \\
\hline
\multirow{2}{*}{Maximum speed} & Less than 25 mph & 2 & 21 & 19 & 42 \\
\hline
& 25 mph or greater & 13 & 47 & 18 & 78 \\
\hline
& Total & 15 & 68 & 37 & 120 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Table 9.2}
\begin{tabular}{|l|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{Expected frequency}} & \multicolumn{3}{|c|}{Bicycle} \\
\hline
& & Road & Gravel & Electric \\
\hline
\multirow{2}{*}{Maximum speed} & Less than 25 mph & & & 12.95 \\
\hline
& 25 mph or greater & & & 24.05 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{enumerate}[label=(\alph*)]
\item Complete the table of expected frequencies in the Printed Answer Booklet.
\item Determine the contribution to the chi-squared test statistic for the Electric bicycle and maximum speed 25 mph or greater. Give your answer correct to 4 decimal places.
The contributions to the chi-squared test statistic for the remaining categories are shown in Table 9.3.
\begin{table}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Table 9.3}
\begin{tabular}{|l|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{Contribution to the test statistic}} & \multicolumn{3}{|c|}{Bicycle} \\
\hline
& & Road & Gravel & Electric \\
\hline
\multirow{2}{*}{Maximum speed} & Less than 25 mph & 2.0119 & 0.3294 & 2.8264 \\
\hline
& 25 mph or greater & 1.0833 & 0.1774 & \\
\hline
\end{tabular}
\end{center}
\end{table}
\item In this question you must show detailed reasoning.
Carry out the test at the 5\% significance level.
\item For each type of bicycle, give a brief interpretation of what the data suggest about maximum speed.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics Major 2024 Q9 [13]}}