| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2021 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Confidence intervals for population mean |
| Difficulty | Moderate -0.8 This is a straightforward confidence interval question requiring standard formulas: calculating sample mean/variance from summations, applying the normal distribution CI formula with z=2.576, and using binomial probability (0.99^4). All steps are routine S2 procedures with no conceptual challenges or novel problem-solving required. |
| Spec | 5.05b Unbiased estimates: of population mean and variance5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\text{est}(\mu) = \frac{7570}{100}\ (= 75.7)\) | B1 | |
| \(\text{est}(\sigma^2) = \frac{100}{99}\left(\frac{\sum h^2}{100} - \text{'75.7'}^2\right)\) or \(\frac{1}{99}\left(588050 - \frac{7570^2}{100}\right) = \frac{100}{99}\left(\frac{588050}{100} - \text{'75.7'}^2\right)\ [= 151.525]\) | M1 | Attempted. (Note: Biased variance (150.01) scores M0) |
| \(= 152\) (3 sf) | A1 | Or \(\frac{15001}{99}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\text{'75.7'} \pm z\sqrt{\frac{\text{'151.525'}}{100}}\) | M1 | For expression of correct form. Must be a \(z\) value. Condone just \(+\) or just \(-\) |
| \(z = 2.576\) | B1 | Accept 2.574 to 2.579 |
| \(72.5\) to \(78.9\) | A1 FT | FT biased variance only. Must be an interval |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(0.99^4\) | B1 | |
| \(0.961\) (3 sf) | B1 |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{est}(\mu) = \frac{7570}{100}\ (= 75.7)$ | B1 | |
| $\text{est}(\sigma^2) = \frac{100}{99}\left(\frac{\sum h^2}{100} - \text{'75.7'}^2\right)$ or $\frac{1}{99}\left(588050 - \frac{7570^2}{100}\right) = \frac{100}{99}\left(\frac{588050}{100} - \text{'75.7'}^2\right)\ [= 151.525]$ | M1 | Attempted. (**Note:** Biased variance (150.01) scores M0) |
| $= 152$ (3 sf) | A1 | Or $\frac{15001}{99}$ |
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{'75.7'} \pm z\sqrt{\frac{\text{'151.525'}}{100}}$ | M1 | For expression of correct form. Must be a $z$ value. Condone just $+$ or just $-$ |
| $z = 2.576$ | B1 | Accept 2.574 to 2.579 |
| $72.5$ to $78.9$ | A1 FT | FT biased variance only. Must be an interval |
## Question 6(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $0.99^4$ | B1 | |
| $0.961$ (3 sf) | B1 | |6 The heights, $h$ centimetres, of a random sample of 100 fully grown animals of a certain species were measured. The results are summarised below.
$$n = 100 \quad \Sigma h = 7570 \quad \Sigma h ^ { 2 } = 588050$$
\begin{enumerate}[label=(\alph*)]
\item Find unbiased estimates of the population mean and variance.
\item Calculate a $99 \%$ confidence interval for the mean height of animals of this species.\\
Four random samples were taken and a $99 \%$ confidence interval for the population mean, $\mu$, was found from each sample.
\item Find the probability that all four of these confidence intervals contain the true value of $\mu$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2021 Q6 [8]}}