| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2019 |
| Session | June |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Reverse-engineering from given variance |
| Difficulty | Moderate -0.8 This is a straightforward application of the variance formula requiring algebraic rearrangement. Students need to recall that s² = [Σx² - (Σx)²/n]/(n-1), substitute the given values (s² = 9.62, Σx² = 4361, n = 50), and solve for Σx to find the mean. It's computational rather than conceptual, with no problem-solving insight required beyond formula manipulation. |
| Spec | 5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\dfrac{50}{49}\left(\dfrac{4361}{50} - \bar{x}^2\right) = 9.62\) | M1 | or \(\left(\dfrac{4361}{49} - \dfrac{(\Sigma x)^2}{50 \times 49}\right) = 9.62\); BOD regarding symbols used |
| \(\bar{x}^2 = \dfrac{4361}{50} - 9.62 \times \dfrac{49}{50} = 77.7924\) | A1 | \((\Sigma x)^2 = 4361 \times 50 - 9.62 \times 50 \times 49 = 194481\) or \(\Sigma x = 441\); \((\Sigma x)\) or \((\bar{x})\) must be correctly identified |
| \(\bar{x} = 8.82\) (3 sf) | A1 | SC use of 'biased' leading to 8.81 B1 |
| Total: 3 |
**Question 2:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{50}{49}\left(\dfrac{4361}{50} - \bar{x}^2\right) = 9.62$ | M1 | or $\left(\dfrac{4361}{49} - \dfrac{(\Sigma x)^2}{50 \times 49}\right) = 9.62$; BOD regarding symbols used |
| $\bar{x}^2 = \dfrac{4361}{50} - 9.62 \times \dfrac{49}{50} = 77.7924$ | A1 | $(\Sigma x)^2 = 4361 \times 50 - 9.62 \times 50 \times 49 = 194481$ or $\Sigma x = 441$; $(\Sigma x)$ or $(\bar{x})$ must be correctly identified |
| $\bar{x} = 8.82$ (3 sf) | A1 | SC use of 'biased' leading to 8.81 B1 |
| **Total: 3** | | |
---2 The length of worms is denoted by $X \mathrm {~cm}$. The lengths of a random sample of 50 worms were measured. Some of the results were lost, but the following results are available.
\begin{itemize}
\item $\Sigma x ^ { 2 } = 4361$
\item An unbiased estimate of the population variance of $X$ is 9.62.
\end{itemize}
Calculate the mean length of the 50 worms.\\
\hfill \mbox{\textit{CAIE S2 2019 Q2 [3]}}