| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2019 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | State distribution of sample mean |
| Difficulty | Easy -1.2 This is a straightforward application of the central limit theorem requiring only recall of standard results: stating that the sample mean has mean μ, standard deviation σ/√n, and is normally distributed. Part (ii) tests basic understanding of when CLT is exact (X itself is normal) versus approximate. No calculations or problem-solving required, just knowledge recall. |
| Spec | 5.05a Sample mean distribution: central limit theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Normal with mean \(372\) | B1 | |
| \(\text{sd} = \dfrac{54}{\sqrt{36}}\) | M1 | or variance \(= \dfrac{54^2}{36}\) M1 |
| \((= 9)\) | A1 | \((= 81)\) A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Pop normal | B1 | Allow \(X\) is normal |
**Question 2:**
**Part (i):**
Normal with mean $372$ | B1 |
$\text{sd} = \dfrac{54}{\sqrt{36}}$ | M1 | or variance $= \dfrac{54^2}{36}$ M1
$(= 9)$ | A1 | $(= 81)$ A1
**Part (ii):**
Pop normal | B1 | Allow $X$ is normal2 The random variable $X$ has mean 372 and standard deviation 54 .\\
(i) Describe fully the distribution of the mean of a random sample of 36 values of $X$.\\
(ii) The distribution in part (i) might be either exact or approximate. State a condition under which the distribution is exact.\\
\hfill \mbox{\textit{CAIE S2 2019 Q2 [4]}}