| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2005 |
| 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: finding the midpoint (simple average), using it with the sample count to find n (one equation to solve), and working backwards from interval width to find the z-value and hence confidence level. All steps are routine applications of S2 formulas with no conceptual challenges, making it slightly easier than average. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \((0.1993 + 0.2887)/2 (= 0.244)\) | B1 | For correct mid-point |
| \(= 61/n\) | M1 | For equating their mid-point with \(61/n\) |
| \(n = 250\) | A1 (3) | For correct answer |
| (ii) \(0.0447 = z \times \sqrt{\frac{0.244(1-0.244)}{250}}\) | M1 | For equating half-width with \(z \times \sqrt{\frac{pq}{n}}\) or equiv. |
| (or equiv. equ. leading to this) | M1 | For solving for \(z\) from a reasonable looking equation |
| \(z = 1.646\) | A1 | For obtaining \(z = 1.64\) or \(1.65\) |
| 90% confidence interval | A1ft (4) | For correct answer (nearest whole no.) |
**(i)** $(0.1993 + 0.2887)/2 (= 0.244)$ | B1 | For correct mid-point
$= 61/n$ | M1 | For equating their mid-point with $61/n$
$n = 250$ | A1 (3) | For correct answer
**(ii)** $0.0447 = z \times \sqrt{\frac{0.244(1-0.244)}{250}}$ | M1 | For equating half-width with $z \times \sqrt{\frac{pq}{n}}$ or equiv.
(or equiv. equ. leading to this) | M1 | For solving for $z$ from a reasonable looking equation
$z = 1.646$ | A1 | For obtaining $z = 1.64$ or $1.65$
90% confidence interval | A1ft (4) | For correct answer (nearest whole no.)3 A survey of a random sample of $n$ people found that 61 of them read The Reporter newspaper. A symmetric confidence interval for the true population proportion, $p$, who read The Reporter is $0.1993 < p < 0.2887$.\\
(i) Find the mid-point of this confidence interval and use this to find the value of $n$.\\
(ii) Find the confidence level of this confidence interval.
\hfill \mbox{\textit{CAIE S2 2005 Q3 [7]}}