| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2017 |
| Session | March |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Justifying CLT for sampling distribution |
| Difficulty | Moderate -0.8 Part (i) is a standard CLT application requiring only the formula for sampling distribution (mean μ, SD σ/√n) and normal probability calculation. Part (ii) tests understanding that CLT is needed because the population distribution of X is not stated to be normal—this is routine conceptual recall rather than problem-solving. The question is straightforward with no novel insight required, making it easier than average. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)5.05a Sample mean distribution: central limit theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{53-52}{6.1\div\sqrt{75}}\) \((= 1.420)\) | M1 | |
| \(\frac{51-52}{6.1\div\sqrt{75}}\) \((= -1.420)\) | M1 | or \(-\)"\(1.420\)" seen |
| \(\Phi(\text{"1.420"}) - \Phi(\text{"-1.420"})\) | M1 | |
| \(= 0.844\) (3 sfs) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Need to assume \(\bar{X}\) (approx.) normally distributed | B1 | or \(X\) not stated to be normally distributed |
## Question 3(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{53-52}{6.1\div\sqrt{75}}$ $(= 1.420)$ | M1 | |
| $\frac{51-52}{6.1\div\sqrt{75}}$ $(= -1.420)$ | M1 | or $-$"$1.420$" seen |
| $\Phi(\text{"1.420"}) - \Phi(\text{"-1.420"})$ | M1 | |
| $= 0.844$ (3 sfs) | A1 | |
## Question 3(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Need to assume $\bar{X}$ (approx.) normally distributed | B1 | or $X$ not stated to be normally distributed |
---3 The length, in centimetres, of a certain type of snake is modelled by the random variable $X$ with mean 52 and standard deviation 6.1. A random sample of 75 snakes is selected, and the sample mean, $\bar { X }$, is found.\\
(i) Find $\mathrm { P } ( 51 < \bar { X } < 53 )$.\\
(ii) Explain why it was necessary to use the Central Limit theorem in the solution to part (i).\\
\hfill \mbox{\textit{CAIE S2 2017 Q3 [5]}}