| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2012 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Calculate CI from summary stats |
| Difficulty | Standard +0.3 Part (i) is a standard confidence interval calculation using known variance (routine z-interval). Part (ii) tests conceptual understanding of confidence intervals—recognizing that by symmetry, P(entire CI below μ) = 0.5% requires understanding the definition but is a single-step insight once grasped. Overall slightly easier than average due to straightforward calculation and a conceptual question with a direct answer. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(z = 2.574\) to \(2.576\) | B1 | |
| \(12.5 \pm z \cdot \frac{3.2}{\sqrt{250}}\) | M1 | Any \(z\), correct form |
| \(12.0\) to \(13.0\) (3 sf) | A1 (3) | Allow 12 to 13 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(0.005\) or \(0.5\%\) | B1 (1) | Not just \(0.5\) |
## Question 1:
### Part (i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $z = 2.574$ to $2.576$ | B1 | |
| $12.5 \pm z \cdot \frac{3.2}{\sqrt{250}}$ | M1 | Any $z$, correct form |
| $12.0$ to $13.0$ (3 sf) | A1 (3) | Allow 12 to 13 |
### Part (ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $0.005$ or $0.5\%$ | B1 (1) | Not just $0.5$ |
---1 Leaves from a certain type of tree have lengths that are distributed with standard deviation 3.2 cm . A random sample of 250 of these leaves is taken and the mean length of this sample is found to be 12.5 cm .\\
(i) Calculate a $99 \%$ confidence interval for the population mean length.\\
(ii) Write down the probability that the whole of a $99 \%$ confidence interval will lie below the population mean.
\hfill \mbox{\textit{CAIE S2 2012 Q1 [4]}}