| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2020 |
| Session | Specimen |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Known variance confidence intervals |
| Difficulty | Moderate -0.5 This is a straightforward confidence interval question requiring standard formulas from the normal distribution. Part (a) involves calculating a 90% CI for the population mean given sample data, and part (b) tests understanding of what a confidence interval represents—both are routine S2 applications with no novel problem-solving required. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(z = 2.576\) | 1, B1 | |
| \(12.5 \pm z \cdot \dfrac{3.2}{\sqrt{250}}\) | 1, M1 | Any \(z\) |
| \(12.0\) to \(13.0\) (3 sf) | 1, A1 | Allow 12 to 13 |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(0.005\) or \(0.5\%\) | 1, B1 | Not just \(0.5\) |
| Total: 1 |
## Question 1(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $z = 2.576$ | 1, B1 | |
| $12.5 \pm z \cdot \dfrac{3.2}{\sqrt{250}}$ | 1, M1 | Any $z$ |
| $12.0$ to $13.0$ (3 sf) | 1, A1 | Allow 12 to 13 |
| **Total: 3** | | |
---
## Question 1(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $0.005$ or $0.5\%$ | 1, B1 | Not just $0.5$ |
| **Total: 1** | | |
---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 .\\
(a) Calculate a 99\% confidence interval for the population mean length.\\
(b) Write down the probability that the whole of a $99 \%$ confidence interval will lie below the population mean.\\
\hfill \mbox{\textit{CAIE S2 2020 Q1 [4]}}