| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2024 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Standard unbiased estimates calculation |
| Difficulty | Moderate -0.8 Part (a) is direct application of standard formulas for unbiased estimates (sample mean and s²). Part (b) requires applying CLT to find a value from the sampling distribution of the mean using normal tables, which is routine S2 content. Both parts are straightforward textbook exercises with no problem-solving or novel insight required. |
| Spec | 5.05a Sample mean distribution: central limit theorem5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\bar{t} = \frac{230}{75}\) [= 3.0666... or 3.07 (3 sf)] [or 46/15] | B1 | |
| \(s^2 = \frac{75}{74}(\frac{930}{75} - (\frac{230}{75})^2)\) or \(\frac{1}{74}(930 - 230^2/75)\) | M1 | Use of correct formula |
| = 3.0360... or 3.04 (3 sf) or = 337/111 | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \([\Phi^{-1}(1 - 0.234)] = 0.726\) | B1 | |
| \(\pm \frac{a - \text{'3.0667'}}{\sqrt{\text{'3.04'}/75}} = \pm\text{'0.726'}\) | 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 |
| Total: 3 |
## Question 3(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\bar{t} = \frac{230}{75}$ [= 3.0666... or 3.07 (3 sf)] [or 46/15] | B1 | |
| $s^2 = \frac{75}{74}(\frac{930}{75} - (\frac{230}{75})^2)$ or $\frac{1}{74}(930 - 230^2/75)$ | M1 | Use of correct formula |
| = 3.0360... or 3.04 (3 sf) or = 337/111 | A1 | |
| **Total: 3** | | |
## Question 3(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $[\Phi^{-1}(1 - 0.234)] = 0.726$ | B1 | |
| $\pm \frac{a - \text{'3.0667'}}{\sqrt{\text{'3.04'}/75}} = \pm\text{'0.726'}$ | 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 |
| **Total: 3** | | |3 The times, $T$ minutes, taken by a random sample of 75 students to complete a test were noted. The results were summarised by $\Sigma t = 230$ and $\Sigma t ^ { 2 } = 930$.
\begin{enumerate}[label=(\alph*)]
\item Calculate unbiased estimates of the population mean and variance of $T$.\\
You should now assume that your estimates from part (a) are the true values of the population mean and variance of $T$.
\item The times taken by another random sample of 75 students were noted, and the sample mean, $\bar { T }$, was found.
Find the value of $a$ such that $P ( \bar { T } > a ) = 0.234$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2024 Q3 [6]}}