| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2014 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Combinations & Selection |
| Type | Describing or comparing sampling methods |
| Difficulty | Easy -4.0 The question text is completely corrupted and unreadable, making it impossible to assess the mathematical difficulty. Cannot provide a valid difficulty rating without being able to understand the actual question content. |
| Spec | 2.03a Mutually exclusive and independent events2.03b Probability diagrams: tree, Venn, sample space |
| Hatchback | Estate | Convertible | |
| Under 25 years | 32 | 11 | 17 |
| Between 25 and 50 years | 45 | 24 | 6 |
| Over 50 years | 31 | 16 | 18 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Expected values: \(32.4\quad 15.3\quad 12.3\) / \(40.5\quad 19.125\quad 15.375\) / \(35.1\quad 16.575\quad 13.325\) | M1 A1 | Find expected values to 1 d.p. |
| \(H_0\): Car type is independent [of age] | B1 | State null hypothesis, A.E.F. |
| \(\chi^2 = 0.005 + 1.208 + 1.796 + 0.500 + 1.243 + 5.716 + 0.479 + 0.020 + 1.640 = 12.6\) (allow \(12.7\)) | M1 A1 | Calculate \(\chi^2\) to 1 d.p. |
| \(\chi^2_{4,\,0.95} = 9.488\) OR \(9.49\) | B1 | State or use correct tabular value |
| Reject \(H_0\) if \(\chi^2 >\) tabular value | M1 | Valid method for conclusion |
| Car type is dependent on age | A1 | Conclusion consistent with correct values, A.E.F. |
## Question 8:
| Answer/Working | Marks | Guidance |
|---|---|---|
| Expected values: $32.4\quad 15.3\quad 12.3$ / $40.5\quad 19.125\quad 15.375$ / $35.1\quad 16.575\quad 13.325$ | M1 A1 | Find expected values to 1 d.p. |
| $H_0$: Car type is independent [of age] | B1 | State null hypothesis, A.E.F. |
| $\chi^2 = 0.005 + 1.208 + 1.796 + 0.500 + 1.243 + 5.716 + 0.479 + 0.020 + 1.640 = 12.6$ (allow $12.7$) | M1 A1 | Calculate $\chi^2$ to 1 d.p. |
| $\chi^2_{4,\,0.95} = 9.488$ OR $9.49$ | B1 | State or use correct tabular value |
| Reject $H_0$ if $\chi^2 >$ tabular value | M1 | Valid method for conclusion |
| Car type is dependent on age | A1 | Conclusion consistent with correct values, A.E.F. |8 A random sample of 200 is taken from the adult population of a town and classified by age-group and preferred type of car. The results are given in the following table.
\begin{center}
\begin{tabular}{ | l | c | c | c | }
\hline
& Hatchback & Estate & Convertible \\
\hline
Under 25 years & 32 & 11 & 17 \\
\hline
Between 25 and 50 years & 45 & 24 & 6 \\
\hline
Over 50 years & 31 & 16 & 18 \\
\hline
\end{tabular}
\end{center}
Test, at the $5 \%$ significance level, whether preferred type of car is independent of age-group.
\hfill \mbox{\textit{CAIE FP2 2014 Q8}}