| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2024 |
| 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 application of standard formulas for unbiased estimates of mean and variance from summary statistics. Part (a) requires direct substitution into memorized formulas (mean = Σx/n, variance = [Σx² - (Σx)²/n]/(n-1)), and part (b) tests basic knowledge that the sample should be random/representative. No problem-solving or conceptual insight required beyond formula recall. |
| Spec | 5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{est}(\mu) = 0.368 = \frac{46}{125}\) | B1 | OE |
| \(\text{est}(\sigma^2) = \frac{100}{99}\left(\frac{17.34}{100} - \text{their } 0.368^2\right)\) or \(\frac{1}{99}\left(17.34 - \frac{36.8^2}{100}\right)\) | M1 | For use of a correct formula (ft *their* \(\mu\)) |
| \(= 0.0384\) (3 sf) | A1 | |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Must be a random sample | B1 | E.g. Values must have been randomly selected; Sample should be representative of the population; All values should have equal chance of being selected; It should be an unbiased sample; Independent sample/insect lengths are independent of one another. ISW |
| 1 |
## Question 1:
**Part (a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{est}(\mu) = 0.368 = \frac{46}{125}$ | **B1** | OE |
| $\text{est}(\sigma^2) = \frac{100}{99}\left(\frac{17.34}{100} - \text{their } 0.368^2\right)$ or $\frac{1}{99}\left(17.34 - \frac{36.8^2}{100}\right)$ | **M1** | For use of a correct formula (ft *their* $\mu$) |
| $= 0.0384$ (3 sf) | **A1** | |
| | **3** | |
**Part (b):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Must be a random sample | **B1** | E.g. Values must have been randomly selected; Sample should be representative of the population; All values should have equal chance of being selected; It should be an unbiased sample; Independent sample/insect lengths are independent of one another. ISW |
| | **1** | |
---1 The lengths, $X \mathrm {~cm}$, of a sample of 100 insects of a certain type were summarised as follows.
$$n = 100 \quad \sum x = 36.8 \quad \sum x ^ { 2 } = 17.34$$
\begin{enumerate}[label=(\alph*)]
\item Calculate unbiased estimates for the population mean and variance of $X$.
\item State a necessary condition for the estimates found in part (a) to be reliable.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2024 Q1 [4]}}