| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2004 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Standard unbiased estimates calculation |
| Difficulty | Moderate -0.8 This is a straightforward application of standard formulas: calculating sample mean and unbiased variance (dividing by n-1), then computing a confidence interval for a proportion using normal approximation. Both parts require only direct substitution into well-rehearsed formulas with no problem-solving or conceptual challenges, making it easier than average. |
| Spec | 5.05b Unbiased estimates: of population mean and variance5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\bar{x} = 375.3\); \(\sigma^2_{n-1} = 8.29\) | B1 M1 A1 | For correct mean (3.s.f); For legit method involving \(n-1\), can be implied; For correct answer |
| 3 | ||
| (ii) \(p = 0.19\) or equiv.; \(0.19 \pm 2.055\sqrt{\frac{0.19 \times 0.81}{200}}\); \(0.133 < p < 0.247\) | B1 M1 B1 A1 | For correct \(p\); For correct form \(p \pm z\sqrt{\frac{pq}{n}}\) either/both sides; For \(z = 2.054\) or \(2.055\); For correct answer |
| 4 |
**(i)** $\bar{x} = 375.3$; $\sigma^2_{n-1} = 8.29$ | B1 M1 A1 | For correct mean (3.s.f); For legit method involving $n-1$, can be implied; For correct answer
| | **3** | |
**(ii)** $p = 0.19$ or equiv.; $0.19 \pm 2.055\sqrt{\frac{0.19 \times 0.81}{200}}$; $0.133 < p < 0.247$ | B1 M1 B1 A1 | For correct $p$; For correct form $p \pm z\sqrt{\frac{pq}{n}}$ either/both sides; For $z = 2.054$ or $2.055$; For correct answer
| | **4** | |4 Packets of cat food are filled by a machine.\\
(i) In a random sample of 10 packets, the weights, in grams, of the packets were as follows.\\
$\begin{array} { l l l l l l l l l l } 374.6 & 377.4 & 376.1 & 379.2 & 371.2 & 375.0 & 372.4 & 378.6 & 377.1 & 371.5 \end{array}$\\
Find unbiased estimates of the population mean and variance.\\
(ii) In a random sample of 200 packets, 38 were found to be underweight. Calculate a $96 \%$ confidence interval for the population proportion of underweight packets.
\hfill \mbox{\textit{CAIE S2 2004 Q4 [7]}}