| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2007 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Calculate CI for proportion |
| Difficulty | Moderate -0.8 This is a straightforward confidence interval question requiring standard formulas for proportion CI and sample size calculation. Part (i) is pure recall, parts (ii) and (iii) are direct applications of bookwork formulas with minimal problem-solving. Easier than average A-level content. |
| Spec | 2.01a Population and sample: terminology5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| (i) A sample where every element has an equal chance of being chosen OR a random sample of size n is a sample chosen in such a way that each possible group of size n has the same chance of being picked. | B1 | 1 |
| (ii) \(130/350(0.371)\) | B1 | For proportion used |
| \(0.371 \pm 1.96 \times \sqrt{\frac{(0.371)(0.629)}{350}}\) | M1 | Correct shape \(\bar{x} \pm z\sqrt{n}\) |
| \(= 0.371 \pm 0.050609 = (0.321, 0.422)\) | B1 | Correct z value 1.96 used |
| A1 | 4 | Correct limits (written as interval) |
| (iii) \(1.96\sqrt{\frac{(0.371)(0.629)}{n}} = 0.02\) | M1* | Seeing an equation involving 0.02 or 0.04, n in denom and a sq rt and proportions used |
| M1^dep | For equation of correct form | |
| \(n = 2241\) or 2242 or 2243 or 2240 | A1 | 3 |
**(i)** A sample where every element has an equal chance of being chosen OR a random sample of size n is a sample chosen in such a way that each possible group of size n has the same chance of being picked. | B1 | 1 |
**(ii)** $130/350(0.371)$ | B1 | For proportion used
$0.371 \pm 1.96 \times \sqrt{\frac{(0.371)(0.629)}{350}}$ | M1 | Correct shape $\bar{x} \pm z\sqrt{n}$
$= 0.371 \pm 0.050609 = (0.321, 0.422)$ | B1 | Correct z value 1.96 used
| A1 | 4 | Correct limits (written as interval)
**(iii)** $1.96\sqrt{\frac{(0.371)(0.629)}{n}} = 0.02$ | M1* | Seeing an equation involving 0.02 or 0.04, n in denom and a sq rt and proportions used
| M1^dep | For equation of correct form
$n = 2241$ or 2242 or 2243 or 2240 | A1 | 3 | Correct whole number answer
---3 (i) Explain what is meant by the term 'random sample'.
In a random sample of 350 food shops it was found that 130 of them had Special Offers.\\
(ii) Calculate an approximate $95 \%$ confidence interval for the proportion of all food shops with Special Offers.\\
(iii) Estimate the size of a random sample required for an approximate $95 \%$ confidence interval for this proportion to have a width of 0.04 .
\hfill \mbox{\textit{CAIE S2 2007 Q3 [8]}}