| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2022 |
| Session | March |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Standard unbiased estimates calculation |
| Difficulty | Easy -1.2 This is a straightforward calculation requiring direct application of standard formulas for sample mean and variance (using n-1 denominator). Part (b) requires basic understanding that small sample size affects reliability. No problem-solving or conceptual depth required—purely procedural with clearly given data. |
| Spec | 2.01b Informal inferences: from samples5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{Est}(\mu) = \frac{1199}{6}\) or \(199.833\) or \(200\) or \(\frac{2398}{12}\) [mm] | B1 | Accept in any form |
| \(\text{Est}(\sigma^2) = \frac{12}{11}\left(\frac{479226}{12} - \frac{'1199'^2}{6}\right)\) or \(\frac{1}{11}\left('479226' - \frac{'2398'^2}{6}\right)\) | M1 | Use of their values in correct formula (may be implied) |
| \(= 2.33\) (3 sf) [mm\(^2\)] | A1 | Accept \(\frac{7}{3}\) |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Small sample | B1 | Accept not 'not representative' unless qualified |
| 1 |
## Question 1:
**Part (a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{Est}(\mu) = \frac{1199}{6}$ or $199.833$ or $200$ or $\frac{2398}{12}$ [mm] | B1 | Accept in any form |
| $\text{Est}(\sigma^2) = \frac{12}{11}\left(\frac{479226}{12} - \frac{'1199'^2}{6}\right)$ or $\frac{1}{11}\left('479226' - \frac{'2398'^2}{6}\right)$ | M1 | Use of their values in correct formula (may be implied) |
| $= 2.33$ (3 sf) [mm$^2$] | A1 | Accept $\frac{7}{3}$ |
| | **3** | |
**Part (b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Small sample | B1 | Accept not 'not representative' unless qualified |
| | **1** | |
---1 The lengths, in millimetres, of a random sample of 12 rods made by a certain machine are as follows.\\
200\\
201\\
198\\
202\\
200\\
199\\
199\\
201\\
197\\
202\\
200\\
199
\begin{enumerate}[label=(\alph*)]
\item Find unbiased estimates of the population mean and variance.
\item Give a statistical reason why these estimates may not be reliable.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2022 Q1 [4]}}