| Exam Board | OCR MEI |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2010 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Interpret association after test |
| Difficulty | Standard +0.3 This is a standard chi-squared test of independence with clearly presented data in a 3×3 contingency table. Students must calculate expected frequencies, compute chi-squared contributions, compare to critical value, and make basic interpretations. While it requires careful calculation across 9 cells, it follows a completely routine procedure taught in S2 with no conceptual challenges or novel elements—making it slightly easier than average for A-level. |
| Spec | 5.06a Chi-squared: contingency tables |
| \multirow{2}{*}{} | Category of runner | \multirow{2}{*}{Row totals} | |||
| Junior | Senior | Veteran | |||
| \multirow{3}{*}{Type of running} | Track | 9 | 8 | 2 | 19 |
| Road | 4 | 8 | 12 | 24 | |
| Both | 4 | 10 | 6 | 20 | |
| Column totals | 17 | 26 | 20 | 63 | |
| Answer | Marks | Guidance |
|---|---|---|
| Part (i) | \(H_0\): no association between category of runner and type of running; \(H_1\): some association between category of runner and type of running; | B1 |
| EXPECTED | Junior | Senior |
| Track | 5.13 | 7.84 |
| Road | 6.48 | 9.90 |
| Both | 5.40 | 8.25 |
| CONTRIBUTN | Junior | Senior |
| Track | 2.9257 | 0.0032 |
| Road | 0.9468 | 0.3663 |
| Both | 0.3615 | 0.3694 |
| \(\chi^2 = 10.21\) | M1 for summation; A1 for \(\chi^2\) | |
| Refer to \(\chi_4^2\) | B1 for 4 deg of f | |
| Critical value at 5% level = 9.488 | B1 CAO for cv | |
| Result is significant | B1 FT their 'sensible' \(\chi^2\) | |
| There is evidence to suggest that there is some association between category of runner and type of running. | E1 must be consistent with their \(\chi^2\) | 4 |
| NB if \(H_0\) \(H_1\) reversed, or 'correlation' mentioned, do not award first B1or final E1 | ||
| Part (ii) | • Juniors appear be track runners more often than expected and road less often than expected. | E1 E1 |
| • Seniors tend to be as expected in all three categories of running. | E1 E1 | |
| • Veterans tend to be road runners more than expected and track runners less than expected. | E1 E1 | |
| TOTAL | 18 |
| **Part (i)** | $H_0$: no association between category of runner and type of running; $H_1$: some association between category of runner and type of running; | B1 | 1 |
|---|---|---|---|
| | **EXPECTED** | Junior | Senior | Veteran |
| | Track | 5.13 | 7.84 | 6.03 |
| | Road | 6.48 | 9.90 | 7.62 |
| | Both | 5.40 | 8.25 | 6.35 | M1 A2 for expected values (to 2 dp) (allow A1 for at least one row or column correct) | 7 |
| | **CONTRIBUTN** | Junior | Senior | Veteran |
| | Track | 2.9257 | 0.0032 | 2.6949 |
| | Road | 0.9468 | 0.3663 | 2.5190 |
| | Both | 0.3615 | 0.3694 | 0.0192 | M1 for valid attempt at $(O-E)^2/E$; A1 for all correct NB These M1A1 marks cannot be implied by a correct final value of $\chi^2$ |
| | $\chi^2 = 10.21$ | M1 for summation; A1 for $\chi^2$ | |
| | Refer to $\chi_4^2$ | B1 for 4 deg of f | |
| | Critical value at 5% level = 9.488 | B1 CAO for cv | |
| | Result is significant | B1 FT their 'sensible' $\chi^2$ | |
| | There is evidence to suggest that there is some association between category of runner and type of running. | E1 must be consistent with their $\chi^2$ | 4 |
| | NB if $H_0$ $H_1$ reversed, or 'correlation' mentioned, do not award first B1or final E1 | | |
| **Part (ii)** | • Juniors appear be track runners more often than expected and road less often than expected. | E1 E1 | 6 |
| | • Seniors tend to be as expected in all three categories of running. | E1 E1 | |
| | • Veterans tend to be road runners more than expected and track runners less than expected. | E1 E1 | |
| | | **TOTAL** | **18** |4 In a survey a random sample of 63 runners is selected. The category of runner and the type of running are classified as follows.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
\multicolumn{2}{|c|}{\multirow{2}{*}{}} & \multicolumn{3}{|c|}{Category of runner} & \multirow{2}{*}{Row totals} \\
\hline
& & Junior & Senior & Veteran & \\
\hline
\multirow{3}{*}{Type of running} & Track & 9 & 8 & 2 & 19 \\
\hline
& Road & 4 & 8 & 12 & 24 \\
\hline
& Both & 4 & 10 & 6 & 20 \\
\hline
\multicolumn{2}{|c|}{Column totals} & 17 & 26 & 20 & 63 \\
\hline
\end{tabular}
\end{center}
(i) Carry out a test at the $5 \%$ significance level to examine whether there is any association between category of runner and the type of running. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic.\\
(ii) For each category of runner, comment briefly on how the type of running compares with what would be expected if there were no association.
\hfill \mbox{\textit{OCR MEI S2 2010 Q4 [18]}}