| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2004 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Confidence intervals for population mean |
| Difficulty | Moderate -0.8 This is a straightforward confidence interval question requiring only standard formulas: calculating sample mean and unbiased variance estimate from summary statistics, then applying the normal distribution confidence interval formula. All steps are routine recall with no problem-solving or conceptual challenges beyond basic substitution into well-practiced formulas. |
| Spec | 5.05b Unbiased estimates: of population mean and variance5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\bar{x} = \frac{4080}{150} = 27.2\) | B1 | For \(4080/150\) |
| \(s^2 = \frac{1}{149}\left(159252 - \frac{4080^2}{150}\right) = 324\) | M1 | For correct expression (from formulae sheet or equiv.) |
| A1 3 | For correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(94\%\ \text{CI} = 27.2 \pm 1.882 \times \sqrt{\frac{324}{150}}\) | M1 | For one of correct form \(\bar{x} + z \times \frac{s}{\sqrt{n}}\) or \(\bar{x} - z \times \frac{s}{\sqrt{n}}\) |
| B1 | For \(z = 1.881\) or \(1.882\) only | |
| A1ft | For correct expression with their \(s/\sqrt{150}\), \(z\) and \(\bar{x}\) | |
| \(= (24.4,\ 30.0)\) | A1 4 | Or equivalent statement (c.w.o.) |
## Question 3(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\bar{x} = \frac{4080}{150} = 27.2$ | B1 | For $4080/150$ |
| $s^2 = \frac{1}{149}\left(159252 - \frac{4080^2}{150}\right) = 324$ | M1 | For correct expression (from formulae sheet or equiv.) |
| | A1 **3** | For correct answer |
## Question 3(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $94\%\ \text{CI} = 27.2 \pm 1.882 \times \sqrt{\frac{324}{150}}$ | M1 | For one of correct form $\bar{x} + z \times \frac{s}{\sqrt{n}}$ or $\bar{x} - z \times \frac{s}{\sqrt{n}}$ |
| | B1 | For $z = 1.881$ or $1.882$ only |
| | A1ft | For correct expression with their $s/\sqrt{150}$, $z$ and $\bar{x}$ |
| $= (24.4,\ 30.0)$ | A1 **4** | Or equivalent statement (c.w.o.) |
---3 A random sample of 150 students attending a college is taken, and their travel times, $t$ minutes, are measured. The data are summarised by $\Sigma t = 4080$ and $\Sigma t ^ { 2 } = 159252$.\\
(i) Calculate unbiased estimates of the population mean and variance.\\
(ii) Calculate a $94 \%$ confidence interval for the population mean travel time.
\hfill \mbox{\textit{CAIE S2 2004 Q3 [7]}}