| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2004 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Justifying CLT for sampling distribution |
| Difficulty | Moderate -0.5 This is a straightforward application of the Central Limit Theorem with a large sample size (n=300). Part (i) requires routine standardization and normal table lookup, while part (ii) simply asks for recall of the CLT condition that normality is not required for large samples. The calculation is mechanical with no conceptual challenges beyond recognizing when CLT applies. |
| Spec | 5.05a Sample mean distribution: central limit theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\bar{X} \sim N\!\left(6.7, \frac{3.1^2}{300}\right)\) | M1 | For standardising (with or without 300 in denom) |
| \(z_1 = \frac{6.8 - 6.7}{3.1/\sqrt{300}} = 0.5587\) | A1 | For two correct expressions for \(z\) |
| \(z_2 = \frac{6.5 - 6.7}{3.1/\sqrt{300}} = -1.117\) | ||
| \(\text{Prob} = \Phi(0.5587) - \{1 - \Phi(1.117)\}\) \(= 0.7119 - (1 - 0.8679)\) | M1 | For subtracting 2 probabilities |
| \(= 0.580\) | A1 4 | For correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| 300 is large, so \(\bar{X}\) is approx normal even if \(X\) is not i.e. CLT application | B1 1 | For reference to large \(n\) and/or CLT |
## Question 2(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\bar{X} \sim N\!\left(6.7, \frac{3.1^2}{300}\right)$ | M1 | For standardising (with or without 300 in denom) |
| $z_1 = \frac{6.8 - 6.7}{3.1/\sqrt{300}} = 0.5587$ | A1 | For two correct expressions for $z$ |
| $z_2 = \frac{6.5 - 6.7}{3.1/\sqrt{300}} = -1.117$ | | |
| $\text{Prob} = \Phi(0.5587) - \{1 - \Phi(1.117)\}$ $= 0.7119 - (1 - 0.8679)$ | M1 | For subtracting 2 probabilities |
| $= 0.580$ | A1 **4** | For correct answer |
## Question 2(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| 300 is large, so $\bar{X}$ is approx normal even if $X$ is not i.e. CLT application | B1 **1** | For reference to large $n$ and/or CLT |
---2 Over a long period of time it is found that the amount of sunshine on any day in a particular town in Spain has mean 6.7 hours and standard deviation 3.1 hours.\\
(i) Find the probability that the mean amount of sunshine over a random sample of 300 days is between 6.5 and 6.8 hours.\\
(ii) Give a reason why it is not necessary to assume that the daily amount of sunshine is normally distributed in order to carry out the calculation in part (i).
\hfill \mbox{\textit{CAIE S2 2004 Q2 [5]}}