| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Justifying CLT for confidence intervals |
| Difficulty | Moderate -0.3 Part (i) is a standard confidence interval calculation with known population standard deviation requiring straightforward application of z-critical values and the formula. Part (ii) tests conceptual understanding of when CLT is needed (large sample size n=180 makes it unnecessary to assume normality), but this is a routine bookwork justification rather than requiring deep insight. |
| Spec | 5.05a Sample mean distribution: central limit theorem5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(z = 1.96\) | B1 | seen |
| \(330.1 \pm z \times \frac{4.8}{\sqrt{180}}\) | M1 | Must be of correct form. Any \(z\) |
| \(= 329.4\) to \(330.8\) (1 dp) | A1 | Must be to 1 dp. Must be an interval. |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Yes, because vol of all cans not stated to be normal | B1 | Or Yes, population not stated to be normal |
| 1 |
## Question 2(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $z = 1.96$ | **B1** | seen |
| $330.1 \pm z \times \frac{4.8}{\sqrt{180}}$ | **M1** | Must be of correct form. Any $z$ |
| $= 329.4$ to $330.8$ (1 dp) | **A1** | Must be to 1 dp. Must be an interval. |
| | **3** | |
---
## Question 2(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Yes, because vol of all cans not stated to be normal | **B1** | Or Yes, population not stated to be normal |
| | **1** | |2 The standard deviation of the volume of drink in cans of Koola is 4.8 centilitres. A random sample of 180 cans is taken and the mean volume of drink in these 180 cans is found to be 330.1 centilitres.\\
(i) Calculate a $95 \%$ confidence interval for the mean volume of drink in all cans of Koola. Give the end-points of your interval correct to 1 decimal place.\\
(ii) Explain whether it was necessary to use the Central Limit theorem in your answer to part (i).\\
\hfill \mbox{\textit{CAIE S2 2018 Q2 [4]}}