| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2006 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Calculate CI for proportion |
| Difficulty | Moderate -0.8 Part (i) is a standard confidence interval calculation for a proportion using the normal approximation—routine application of a formula with given values. Part (ii) requires identifying basic sampling bias (e.g., non-random sample, time/location bias), which is elementary statistical reasoning. Both parts are straightforward recall and application with no problem-solving complexity. |
| Spec | 5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(p = \frac{203}{278} (= 0.7302 \approx 0.73)\) | B1 | Correct \(p\) |
| \(0.7302 \pm 2.17 \times \sqrt{\frac{pq}{n}} = 0.7302 \pm 2.17 \times \sqrt{\frac{0.7302 \times 0.2698}{278}}\) | M1 | Correct form \(p \pm z \times \sqrt{\frac{pq}{n}}\) either/both sides |
| Correct \(z\) | B1 | |
| \(= (0.672, 0.788)\) | A1 | Correct answer |
| (ii) Mainly unemployed, retired, or mothers with children ie not representative of whole pop | B1 | Or any sensible equivalent |
**(i)** $p = \frac{203}{278} (= 0.7302 \approx 0.73)$ | B1 | Correct $p$
$0.7302 \pm 2.17 \times \sqrt{\frac{pq}{n}} = 0.7302 \pm 2.17 \times \sqrt{\frac{0.7302 \times 0.2698}{278}}$ | M1 | Correct form $p \pm z \times \sqrt{\frac{pq}{n}}$ either/both sides
Correct $z$ | B1 |
$= (0.672, 0.788)$ | A1 | Correct answer | 4 marks total
**(ii)** Mainly unemployed, retired, or mothers with children ie not representative of whole pop | B1 | Or any sensible equivalent | 1 mark
---3 A survey was conducted to find the proportion of people owning DVD players. It was found that 203 out of a random sample of 278 people owned a DVD player.\\
(i) Calculate a $97 \%$ confidence interval for the true proportion of people who own a DVD player.
A second survey to find the proportion of people owning DVD players was conducted at 10 o'clock on a Thursday morning in a shopping centre.\\
(ii) Give one reason why this is not a satisfactory sample.
\hfill \mbox{\textit{CAIE S2 2006 Q3 [5]}}