| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2024 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Central limit theorem |
| Type | Unbiased estimator from summary statistics |
| Difficulty | Standard +0.3 This is a straightforward application of standard formulas: part (a) uses unbiased estimator formulas (routine calculation), and part (b) applies the Central Limit Theorem with a normal distribution lookup. Both parts are direct textbook exercises requiring no problem-solving insight, though slightly above average difficulty due to the CLT application. |
| Spec | 5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks |
|---|---|
| 3(a) | t = 230 [= 3.0666… or 3.07 (3 sf)] [ 0r 46/15 ] |
| 75 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| 74 75 75 | M1 | Use of correct formula. |
| = 3.0360… or 3.04 (3 sf) or = 337/111 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 3(b) | [ Φ−1(1 − 0.234) ] = 0.726 | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| '3.04'/75 | M1 | Ft their 0.726 but must be a z value. |
| Answer | Marks | Guidance |
|---|---|---|
| a = 3.21 (3 sf) | A1 | CWO |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 3:
--- 3(a) ---
3(a) | t = 230 [= 3.0666… or 3.07 (3 sf)] [ 0r 46/15 ]
75 | B1
s2 = 75(930−(230)2) or 1/74(930 – 2302/75 )
74 75 75 | M1 | Use of correct formula.
= 3.0360… or 3.04 (3 sf) or = 337/111 | A1
3
--- 3(b) ---
3(b) | [ Φ−1(1 − 0.234) ] = 0.726 | B1
a−'3.0667'
± = ± ‘0.726’
'3.04'/75 | M1 | Ft their 0.726 but must be a z value.
Note using 0.766 is M0.
Must have sqrt 75.
a = 3.21 (3 sf) | A1 | CWO
3
Question | Answer | Marks | GuidanceThe times, $T$ minutes, taken by a random sample of $75$ students to complete a test were noted. The results were summarised by $\sum t = 230$ and $\sum t^2 = 930$.
\begin{enumerate}[label=(\alph*)]
\item Calculate unbiased estimates of the population mean and variance of $T$. [3]
\end{enumerate}
You should now assume that your estimates from part (a) are the true values of the population mean and variance of $T$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item The times taken by another random sample of $75$ students were noted, and the sample mean, $\overline{T}$, was found.
Find the value of $a$ such that $P(\overline{T} > a) = 0.234$. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2024 Q3 [6]}}