| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2021 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Justifying CLT for sampling distribution |
| Difficulty | Moderate -0.8 This is a straightforward application of sampling distribution of the mean with a normal population. Part (a) requires only calculating the standard error (σ/√n) and finding a probability from the normal distribution—routine mechanics with no conceptual challenge. Part (b) tests basic understanding that CLT is unnecessary when the population is already normal. Both parts are simpler than average A-level questions as they require minimal problem-solving and test only fundamental concepts. |
| Spec | 5.04b Linear combinations: of normal distributions5.05a Sample mean distribution: central limit theorem |
| Answer | Marks |
|---|---|
| 2(a) | 123−125 |
| Answer | Marks | Guidance |
|---|---|---|
| 40 | M1 | Must have √40 |
| Answer | Marks | Guidance |
|---|---|---|
| P(z < ‘–2.108’) = 1 – Φ(‘2.108’) | M1 | For correct probability area consistent with their |
| Answer | Marks |
|---|---|
| = 0.0175 or 0.0176 (3 sf) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 2(b) | No, population is normal | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 2:
--- 2(a) ---
2(a) | 123−125
± [= –2.108...]
6
40 | M1 | Must have √40
No standard deviation/variance mix. Ignore any
continuity correction attempts for this mark.
P(z < ‘–2.108’) = 1 – Φ(‘2.108’) | M1 | For correct probability area consistent with their
working.
= 0.0175 or 0.0176 (3 sf) | A1
3
--- 2(b) ---
2(b) | No, population is normal | B1 | Need both.
1
Question | Answer | Marks | GuidanceThe time, in minutes, taken by students to complete a test has the distribution $\text{N}(125, 36)$.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that the mean time taken to complete the test by a random sample of 40 students is less than 123 minutes. [3]
\item Explain whether it was necessary to use the Central Limit theorem in the solution to part (a). [1]
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2021 Q2 [4]}}