| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2009 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Recover sample stats from CI |
| Difficulty | Standard +0.3 This is a straightforward confidence interval question requiring standard formulas. Part (i) involves simple arithmetic to find the midpoint and use the width formula to find σ². Part (ii) requires rearranging the confidence interval width formula to solve for n. While it involves multiple steps, all techniques are routine applications of standard S2 content 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 |
|---|---|---|
| (i) \(\bar{\mu} = 227.(1)\) | B1, B1 | Correct mean; 2.17 seen |
| \(5 = 2.17 \times \sqrt{\frac{\hat{\sigma}^2}{50}}\) | M1 | Solving an equation with 5 or 10 on the LHS and some \(z\) value \(\times \frac{\hat{\sigma}}{\sqrt{n}}\) on the RHS |
| \(\hat{\sigma}^2 = 265\) or \(266\) | A1 | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) \(4 = 2.17 \times \frac{16.3}{\sqrt{n}}\) | B1ft | Correct equation fit their wrong \(z\) if the same as in part (i) and their \(\hat{\sigma}\) |
| \(n = 78\) | M1, A1 | Solving an equation with their \(z\) and \(\hat{\sigma}\), and width 4 or 8; Correct answer (whole number) |
(i) $\bar{\mu} = 227.(1)$ | B1, B1 | Correct mean; 2.17 seen
$5 = 2.17 \times \sqrt{\frac{\hat{\sigma}^2}{50}}$ | M1 | Solving an equation with 5 or 10 on the LHS and some $z$ value $\times \frac{\hat{\sigma}}{\sqrt{n}}$ on the RHS
$\hat{\sigma}^2 = 265$ or $266$ | A1 | Correct answer
**[4]**
(ii) $4 = 2.17 \times \frac{16.3}{\sqrt{n}}$ | B1ft | Correct equation fit their wrong $z$ if the same as in part (i) and their $\hat{\sigma}$
$n = 78$ | M1, A1 | Solving an equation with their $z$ and $\hat{\sigma}$, and width 4 or 8; Correct answer (whole number)
**[3]**2 The weights in grams of oranges grown in a certain area are normally distributed with mean $\mu$ and standard deviation $\sigma$. A random sample of 50 of these oranges was taken, and a $97 \%$ confidence interval for $\mu$ based on this sample was (222.1, 232.1).\\
(i) Calculate unbiased estimates of $\mu$ and $\sigma ^ { 2 }$.\\
(ii) Estimate the sample size that would be required in order for a $97 \%$ confidence interval for $\mu$ to have width 8 .
\hfill \mbox{\textit{CAIE S2 2009 Q2 [7]}}