| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2022 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Unbiased estimates then CI |
| Difficulty | Moderate -0.8 Part (a) is a standard confidence interval calculation requiring only formula application with given summary statistics. Part (b) tests direct understanding of confidence interval interpretation (95% means 95% of intervals contain the true mean, so 0.95 × 40 = 38), which is conceptual but straightforward recall of definition. No complex problem-solving or novel insight required. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\text{Est}(\mu) = \frac{2520}{200}\) \([= 12.6]\) | B1 | OE |
| \(\text{Est}(\sigma^2) = \frac{200}{199}\left(\frac{31582}{200} - 12.6^2\right)\) or \(\frac{1}{199}\left(31582 - \frac{2520^2}{200}\right)\) | M1 | Allow M1 if \(\frac{200}{199}\) omitted |
| \(= 0.5025\) or \(0.503\) or \(\frac{100}{199}\) | A1 | CWO or \(\sigma = 0.7088\) or \(0.709\) |
| \(z = 1.96\) | B1 | |
| \(12.6 \pm z \times \sqrt{0.5025 \div 200}\) | M1 | For expression of correct form. Any \(z\) but must be \(z\) |
| \(\text{CI} = 12.5\) to \(12.7\) (3 sf) | A1 | CWO. Must be an interval. Note: Use of biased can score maximum B1 M1 A0 B1 M1 A0 |
| 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(0.95 \times 40\ [= 38]\) | B1 | Give at early stage |
| 1 |
## Question 1(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{Est}(\mu) = \frac{2520}{200}$ $[= 12.6]$ | B1 | OE |
| $\text{Est}(\sigma^2) = \frac{200}{199}\left(\frac{31582}{200} - 12.6^2\right)$ or $\frac{1}{199}\left(31582 - \frac{2520^2}{200}\right)$ | M1 | Allow M1 if $\frac{200}{199}$ omitted |
| $= 0.5025$ or $0.503$ or $\frac{100}{199}$ | A1 | CWO or $\sigma = 0.7088$ or $0.709$ |
| $z = 1.96$ | B1 | |
| $12.6 \pm z \times \sqrt{0.5025 \div 200}$ | M1 | For expression of correct form. Any $z$ but must be $z$ |
| $\text{CI} = 12.5$ to $12.7$ (3 sf) | A1 | CWO. Must be an interval. Note: Use of biased can score maximum B1 M1 A0 B1 M1 A0 |
| | **6** | |
---
## Question 1(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $0.95 \times 40\ [= 38]$ | B1 | Give at early stage |
| | **1** | |1 The diameters, $x$ millimetres, of a random sample of 200 discs made by a certain machine were recorded. The results are summarised below.
$$n = 200 \quad \Sigma x = 2520 \quad \Sigma x ^ { 2 } = 31852$$
\begin{enumerate}[label=(\alph*)]
\item Calculate a 95\% confidence interval for the population mean diameter.
\item Jean chose 40 random samples and used each sample to calculate a 95\% confidence interval for the population mean diameter.
How many of these 40 confidence intervals would be expected to include the true value of the population mean diameter?
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2022 Q1 [7]}}