| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared test of independence |
| Type | Standard 2×3 contingency table |
| Difficulty | Standard +0.3 This is a standard chi-squared test of independence with a 2×3 contingency table. Students must calculate expected frequencies, compute the test statistic using the standard formula, find degrees of freedom (2), and compare to critical value. While it requires multiple computational steps, it follows a completely routine procedure with no conceptual challenges or novel insights required—making it slightly easier than average. |
| Spec | 5.06a Chi-squared: contingency tables |
| Driving school attended | |||
| \(A\) | \(B\) | \(C\) | |
| Passes | 23 | 15 | 17 |
| Failures | 27 | 25 | 43 |
| Answer | Marks |
|---|---|
| 8 | Find expected values to (at least) 1 dp: A B C |
| Answer | Marks | Guidance |
|---|---|---|
| Correct conclusion (A.E.F., requires correct values): No association A1 | 7 | [7] |
Question 8:
8 | Find expected values to (at least) 1 dp: A B C
(lose A1 if one or more errors Passes 18⋅33 14⋅67 22⋅00
or if rounded to integers) Failures 31⋅67 25⋅33 38⋅00 M1 A1
State (at least) null hypothesis (A.E.F.): H0: Test result indep of school B1
Calculate value of χ2 : χ2 = 3⋅7 ± 0⋅02 B1
S.R. If rounded to integers above allow: χ2 = 3⋅96 or 4⋅0 (earns max 6/7) (B1)
Compare with tabular value (to 2 dp): χ 2, 0.95 2 = 5⋅99 B1
χ2
Valid method for reaching conclusion: Reject H0 if > tabular value M1
Correct conclusion (A.E.F., requires correct values): No association A1 | 7 | [7]The owner of three driving schools, $A$, $B$ and $C$, wished to assess whether there was an association between passing the driving test and the school attended. He selected a random sample of learner drivers from each of his schools and recorded the numbers of passes and failures at each school. The results that he obtained are shown in the table below.
\begin{center}
\begin{tabular}{|l|c|c|c|}
\hline
& \multicolumn{3}{|c|}{Driving school attended} \\
\hline
& $A$ & $B$ & $C$ \\
\hline
Passes & 23 & 15 & 17 \\
\hline
Failures & 27 & 25 & 43 \\
\hline
\end{tabular}
\end{center}
Using a $\chi^2$-test and a 5% level of significance, test whether there is an association between passing or failing the driving test and the driving school attended. [7]
\hfill \mbox{\textit{CAIE FP2 2010 Q8 [7]}}