| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Calculate CI from summary stats |
| Difficulty | Moderate -0.8 This is a straightforward confidence interval question requiring standard formula application (part i) and understanding that larger sample size reduces interval width (part ii). Both parts involve routine recall and basic conceptual understanding with no problem-solving or novel insight required. |
| Spec | 5.05a Sample mean distribution: central limit theorem5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| \(52 \pm z \times \frac{6.5}{\sqrt{15}}\) | M1 | Expression of the correct form. Any \(z\) |
| \(z = 1.96\) | B1 | Seen or used |
| \(48.7\) to \(55.3\) (3 sf) | A1 | Must be an interval |
| Answer | Marks | Guidance |
|---|---|---|
| Narrower because more information or because \(\frac{\sigma}{\sqrt{n}}\) smaller | B1 | Accept 'sample size is larger', 'more employees', 'width inversely proportional to sq root of n', 'if n increases width decreases', '95% CI is 49.7 to 54.3' or similar. No contradictions |
**Question 3(i):**
$52 \pm z \times \frac{6.5}{\sqrt{15}}$ | M1 | Expression of the correct form. Any $z$
$z = 1.96$ | B1 | Seen or used
$48.7$ to $55.3$ (3 sf) | A1 | Must be an interval
**Total: 3**
## Question 3(ii):
| Narrower because more information or because $\frac{\sigma}{\sqrt{n}}$ smaller | B1 | Accept 'sample size is larger', 'more employees', 'width inversely proportional to sq root of n', 'if n increases width decreases', '95% CI is 49.7 to 54.3' or similar. No contradictions |
---3 The management of a factory wished to find a range within which the time taken to complete a particular task generally lies. It is given that the times, in minutes, have a normal distribution with mean $\mu$ and standard deviation 6.5. A random sample of 15 employees was chosen and the mean time taken by these employees was found to be 52 minutes.\\
(i) Calculate a $95 \%$ confidence interval for $\mu$.\\
Later another $95 \%$ confidence interval for $\mu$ was found, based on a random sample of 30 employees.\\
(ii) State, with a reason, whether the width of this confidence interval was less than, equal to or greater than the width of the previous interval.\\
\hfill \mbox{\textit{CAIE S2 2018 Q3 [4]}}