| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2005 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Calculate CI from summary stats |
| Difficulty | Moderate -0.3 This is a straightforward confidence interval question requiring standard formula application (z-interval for known variance) and interpretation. Part (i) is trivial recall, part (ii)(a) involves routine calculation with given values, and part (ii)(b) asks for basic interpretation comparing the interval to a given value. While it requires understanding of hypothesis testing concepts, it involves no problem-solving or novel insight—slightly easier than average due to its direct nature. |
| Spec | 2.01a Population and sample: terminology5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| (i) for example cheaper, less time consuming, not all destructive | B1 (1 mark) | Or any other legit reason |
| (ii) (a) \(69.3 \pm 1 = 1.645 \times \frac{8.1}{\sqrt{110}} = (68.0, 70.6)\) | M1, B1, A1, A1 ft (4 marks) | For correct form ie \(\bar{x} \pm z\sqrt{n}\); For 1.645; For correct answer; Or equivalent, if on their limits |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) (b) 71.2 not in CI. Sig diff in life span from national average | B1, B1 (2 marks) | Need to see 'life span' and 'difference' |
**(i)** for example cheaper, less time consuming, not all destructive | B1 (1 mark) | Or any other legit reason
**(ii) (a)** $69.3 \pm 1 = 1.645 \times \frac{8.1}{\sqrt{110}} = (68.0, 70.6)$ | M1, B1, A1, A1 ft (4 marks) | For correct form ie $\bar{x} \pm z\sqrt{n}$; For 1.645; For correct answer; Or equivalent, if on their limits
We are 90% confident that the true mean lies between 68.0 and 70.6
**(ii) (b)** 71.2 not in CI. Sig diff in life span from national average | B1, B1 (2 marks) | Need to see 'life span' and 'difference'
---4 (i) Give a reason why, in carrying out a statistical investigation, a sample rather than a complete population may be used.\\
(ii) Rose wishes to investigate whether men in her town have a different life-span from the national average of 71.2 years. She looks at government records for her town and takes a random sample of the ages of 110 men who have died recently. Their mean age in years was 69.3 and the unbiased estimate of the population variance was 65.61.
\begin{enumerate}[label=(\alph*)]
\item Calculate a $90 \%$ confidence interval for the population mean and explain what you understand by this confidence interval.
\item State with a reason what conclusion about the life-span of men in her town Rose could draw from this confidence interval.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2005 Q4 [7]}}