| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2017 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Sample size determination |
| Difficulty | Standard +0.3 This is a straightforward confidence interval question requiring standard formulas. Part (i) involves direct substitution into the CI formula with given values. Part (ii) requires working backwards from the interval width to find n, which is slightly less routine but still a standard textbook exercise with no novel insight required. |
| Spec | 5.05a Sample mean distribution: central limit theorem5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(4820 \pm z \times \dfrac{1420}{\sqrt{125}}\) | M1 | Must be a \(z\) value |
| \(z = 2.326\) | B1 | Accept 2.326 – 2.329 |
| 4524/4525 to 5115/5116 or 4520 to 5120 (3 sf) | A1 | Must be an interval |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\bar{x} = 4840\) | B1 | or width \(= 280\) or half width \(= 140\) |
| \(4840 + 1.96 \times \dfrac{1420}{\sqrt{n}} = 4980\) OE | M1 | or \(140 = 1.96 \times \dfrac{1420}{\sqrt{n}}\) OE |
| \(n = 395\) | A1 | CAO must be an integer |
| Total: 3 |
## Question 2:
**Part 2(i):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $4820 \pm z \times \dfrac{1420}{\sqrt{125}}$ | M1 | Must be a $z$ value |
| $z = 2.326$ | B1 | Accept 2.326 – 2.329 |
| 4524/4525 to 5115/5116 or 4520 to 5120 (3 sf) | A1 | Must be an interval |
| | **Total: 3** | |
**Part 2(ii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\bar{x} = 4840$ | B1 | or width $= 280$ or half width $= 140$ |
| $4840 + 1.96 \times \dfrac{1420}{\sqrt{n}} = 4980$ OE | M1 | or $140 = 1.96 \times \dfrac{1420}{\sqrt{n}}$ OE |
| $n = 395$ | A1 | CAO must be an integer |
| | **Total: 3** | |
---2 The number of words in History essays by students at a certain college has mean $\mu$ and standard deviation 1420.\\
(i) The mean number of words in a random sample of 125 History essays was found to be 4820 . Calculate a $98 \%$ confidence interval for $\mu$.\\
(ii) Another random sample of $n$ History essays was taken. Using this sample, a $95 \%$ confidence interval for $\mu$ was found to be 4700 to 4980 , both correct to the nearest integer. Find the value of $n$.\\
\hfill \mbox{\textit{CAIE S2 2017 Q2 [6]}}