| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2019 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Recover sample stats from CI |
| Difficulty | Challenging +1.2 This question requires understanding of confidence interval formulas and algebraic manipulation to work backwards from a given CI to find sample statistics. While it involves multiple steps and careful algebra with the t-distribution formula, the concepts are standard A-level further maths material with no novel problem-solving required. The backwards working adds moderate difficulty beyond routine CI calculation. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| 6(i) | t √(s2/9) = ½ (1⋅85 – 1⋅65) [= 0⋅1] | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| 8, 0.975 | A1 | Use of correct tabular t-value |
| s2 = 9 × 0⋅043372 = 0⋅0169 or 0⋅130[1]2 | A1 |
| Answer | Marks |
|---|---|
| 6(ii) | x = 1 (1⋅65 + 1⋅85) = 1⋅75 |
| Answer | Marks | Guidance |
|---|---|---|
| or Σ x = 9 × 1⋅75 = 15⋅75 | M1 A1 | Find sample meanx |
| Answer | Marks | Guidance |
|---|---|---|
| or {Σ x2 – (Σ x)2 / 9} / 8 | M1 | or Σ x |
| Σ x2 = 8 × 0⋅0169 + 15⋅752/9 = 27⋅7 | A1 | Find Σ x2 from s2 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 6:
--- 6(i) ---
6(i) | t √(s2/9) = ½ (1⋅85 – 1⋅65) [= 0⋅1] | M1 | Find estimate s2 of population variance (must be t)
t = 2⋅306 (to 3 s.f.)
8, 0.975 | A1 | Use of correct tabular t-value
s2 = 9 × 0⋅043372 = 0⋅0169 or 0⋅130[1]2 | A1
3
--- 6(ii) ---
6(ii) | x = 1 (1⋅65 + 1⋅85) = 1⋅75
2
or Σ x = 9 × 1⋅75 = 15⋅75 | M1 A1 | Find sample meanx
s2 = (Σ x2 – 9 ×x2) / 8
or {Σ x2 – (Σ x)2 / 9} / 8 | M1 | or Σ x
Σ x2 = 8 × 0⋅0169 + 15⋅752/9 = 27⋅7 | A1 | Find Σ x2 from s2
4
Question | Answer | Marks | GuidanceA random sample of 9 members is taken from the large number of members of a sports club, and their heights are measured. The heights of all the members of the club are assumed to be normally distributed. A 95% confidence interval for the population mean height, $\mu$ metres, is calculated from the data as $1.65 \leqslant \mu \leqslant 1.85$.
\begin{enumerate}[label=(\roman*)]
\item Find an unbiased estimate for the population variance.
[3]
\item Denoting the height of a member of the club by $x$ metres, find $\Sigma x^2$ for this sample of 9 members.
[4]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP2 2019 Q6 [7]}}