| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2014 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Estimated variance confidence interval |
| Difficulty | Moderate -0.3 This is a straightforward confidence interval question requiring standard formulas for sample mean and variance, then applying the normal distribution. The CLT part (ii) tests conceptual understanding but is basic. Slightly easier than average as it's purely procedural with no problem-solving or novel insight required. |
| Spec | 5.05a Sample mean distribution: central limit theorem5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Part (i) | ||
| \(\text{Est}(\mu) = \frac{7220}{80}\) or 90.25 | B1 | Accept 90.3 |
| \(\text{Est}(\sigma^2) = \frac{80}{79}\left(\frac{656060}{80} - \left(\frac{7220}{80}\right)^2\right)\) | M1 | \(\frac{1}{79}\left(656060 - \frac{7220^2}{80}\right)\) |
| \(= 56.3924\) or \(\frac{4455}{79}\) | A1 | Accept 56.4 |
| \(z = 2.17\) | B1 | |
| \(\frac{7220}{80} \pm z\sqrt{\frac{'56.3924'}{80}}\) | M1 | Expression of correct form |
| \(= 88.4\) to 92.1 (3 sf) | A1 | [6] Must be an interval (N.B. biased var gives 88.4 to 92.1 scores possible B1M0A0B1M1A1) |
| Part (ii) | ||
| Pop normal | B1 | \(X\) normal or full definition of pop normal |
| No | B1dep | [2] SR B1 for "no" and relevant reference to normal |
| Total | [8] |
| **Part (i)** |
|---|
| $\text{Est}(\mu) = \frac{7220}{80}$ or 90.25 | B1 | Accept 90.3 |
| $\text{Est}(\sigma^2) = \frac{80}{79}\left(\frac{656060}{80} - \left(\frac{7220}{80}\right)^2\right)$ | M1 | $\frac{1}{79}\left(656060 - \frac{7220^2}{80}\right)$ |
| $= 56.3924$ or $\frac{4455}{79}$ | A1 | Accept 56.4 |
| $z = 2.17$ | B1 | |
| $\frac{7220}{80} \pm z\sqrt{\frac{'56.3924'}{80}}$ | M1 | Expression of correct form |
| $= 88.4$ to 92.1 (3 sf) | A1 | [6] Must be an interval (N.B. biased var gives 88.4 to 92.1 scores possible B1M0A0B1M1A1) |
| **Part (ii)** |
|---|
| Pop normal | B1 | $X$ normal or full definition of pop normal |
| No | B1dep | [2] SR B1 for "no" and relevant reference to normal |
| **Total** | [8] |3 The times, in minutes, taken by people to complete a walk are normally distributed with mean $\mu$. The times, $t$ minutes, for a random sample of 80 people were summarised as follows.
$$\Sigma t = 7220 \quad \Sigma t ^ { 2 } = 656060$$
(i) Calculate a $97 \%$ confidence interval for $\mu$.\\
(ii) Explain whether it was necessary to use the Central Limit theorem in part (i).
\hfill \mbox{\textit{CAIE S2 2014 Q3 [8]}}