| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2012 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Single sample confidence interval t-distribution |
| Difficulty | Standard +0.3 This is a straightforward application of t-distribution confidence intervals with standard formulas. Part (i) requires routine calculation from summary statistics, and part (ii) tests conceptual understanding of how sample size and variance affect interval width—both are textbook exercises requiring no novel insight or complex multi-step reasoning. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| State suitable assumption: Population is Normal | B1 | (A.E.F.) |
| Find confidence interval: \(1110.8/10 \pm t\sqrt{(333.9/90)}\) | M1 A1 | |
| \(= 111.1 \pm t\sqrt{3.71}\) | A1 | |
| State/use correct tabular value of \(t\): \(t_{9, 0.995} = 3.25\) | A1 | |
| Evaluate C.I.: \(111 \pm 6\) or \([105, 117]\) | A1 | Total part: [6] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Compare \(t\), estimated variance \(s\) and \(n\): \(t\) and \(s\) smaller, \(n\) larger | M1 | |
| Deduce effect on width of C.I.: Width is less than in (i) | A1 | Total part: [2], Question total: [8] |
| S.R. B1 if valid apart from considering \(n\) |
## Question 7(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| State suitable assumption: Population is Normal | B1 | (A.E.F.) |
| Find confidence interval: $1110.8/10 \pm t\sqrt{(333.9/90)}$ | M1 A1 | |
| $= 111.1 \pm t\sqrt{3.71}$ | A1 | |
| State/use correct tabular value of $t$: $t_{9, 0.995} = 3.25$ | A1 | |
| Evaluate C.I.: $111 \pm 6$ or $[105, 117]$ | A1 | Total part: [6] |
## Question 7(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Compare $t$, estimated variance $s$ and $n$: $t$ and $s$ smaller, $n$ larger | M1 | |
| Deduce effect on width of C.I.: Width is less than in (i) | A1 | Total part: [2], Question total: [8] |
| **S.R.** B1 if valid apart from considering $n$ | | |
---7 The speed $v$ at which a javelin is thrown by an athlete is measured in $\mathrm { km } \mathrm { h } ^ { - 1 }$. The results for 10 randomly chosen throws are summarised by
$$\Sigma v = 1110.8 , \quad \Sigma ( v - \bar { v } ) ^ { 2 } = 333.9$$
where $\bar { v }$ is the sample mean.\\
(i) Stating any necessary assumption, calculate a $99 \%$ confidence interval for the mean speed of a throw.
The results for a further 5 randomly chosen throws are now combined with the above results. It is found that the sample variance is smaller than that used in part (i).\\
(ii) State, with reasons, whether a $95 \%$ confidence interval calculated from the combined 15 results will be wider or less wide than that found in part (i).
\hfill \mbox{\textit{CAIE FP2 2012 Q7 [8]}}