| Exam Board | AQA |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2008 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Proportion confidence interval |
| Difficulty | Moderate -0.3 This is a straightforward application of the normal approximation for a proportion confidence interval with standard interpretation. While it requires knowledge of the CLT and z-values, it's a routine S3 procedure with no conceptual challenges—slightly easier than average due to its mechanical nature and clear two-part structure. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Part (a) | ||
| \(\hat{p} = \frac{132}{200} = 0.66\) | B1 | CAO; or equivalent |
| \(98\% \Rightarrow z = 2.32 \text{ to } 2.33\) | B1 | AWFW (2.3263) |
| CI for \(p\): \(\hat{p} \pm z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\) | M1 | Variance term |
| CI expression used | M1 | |
| ie \(0.66 \pm 2.3263 \times \sqrt{\frac{0.66 \times 0.34}{200}}\) | A1↑ | ft on \(\hat{p}\) and \(z\) |
| ie \(0.66 \pm 0.08\) or \((0.58, 0.74)\) | A1 | AWRT; or equivalent; 6 marks |
| Part (b) | ||
| Value of 0.6 (60%) is within CI | B1↑ | ft on (a) |
| Reason to doubt claim of more than 60% | B1↑ | dependent on previous B1 ft on (a); or equivalent; 2 marks |
| **Part (a)** |
|---|
| $\hat{p} = \frac{132}{200} = 0.66$ | B1 | CAO; or equivalent |
| $98\% \Rightarrow z = 2.32 \text{ to } 2.33$ | B1 | AWFW (2.3263) |
| CI for $p$: $\hat{p} \pm z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ | M1 | Variance term |
| CI expression used | M1 | |
| ie $0.66 \pm 2.3263 \times \sqrt{\frac{0.66 \times 0.34}{200}}$ | A1↑ | ft on $\hat{p}$ and $z$ |
| ie $0.66 \pm 0.08$ or $(0.58, 0.74)$ | A1 | AWRT; or equivalent; 6 marks |
| **Part (b)** |
|---|
| Value of 0.6 (60%) is within CI | B1↑ | ft on (a) |
| Reason to doubt claim of more than 60% | B1↑ | dependent on previous B1 ft on (a); or equivalent; 2 marks |
**Total: 8 marks**
---2 A survey of a random sample of 200 passengers on UK internal flights revealed that 132 of them were on business trips.
\begin{enumerate}[label=(\alph*)]
\item Construct an approximate $98 \%$ confidence interval for the proportion of passengers on UK internal flights that are on business trips.
\item Hence comment on the claim that more than 60 per cent of passengers on UK internal flights are on business trips.
\end{enumerate}
\hfill \mbox{\textit{AQA S3 2008 Q2 [8]}}