| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2019 |
| Session | March |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Standard unbiased estimates calculation |
| Difficulty | Moderate -0.8 This is a straightforward application of standard formulas for unbiased estimates (mean = Σx/n, variance = Σx²/(n-1) - n(x̄)²/(n-1)) followed by a routine normal distribution calculation using the Central Limit Theorem. All steps are mechanical with no problem-solving or insight required—easier than average A-level statistics questions. |
| Spec | 5.05a Sample mean distribution: central limit theorem5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\bar{x} = \frac{420}{50} = 8.4\) | B1 | |
| \(s^2 = \frac{50}{49}\left(\frac{27530}{50} - \left(\frac{420}{50}\right)^2\right)\) | M1 | Or \(\frac{1}{49}(27530 - (420)^2/50)\) |
| \(= 489.8(36...)\) | A1 | Must see \(\geqslant 4\) sf |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\Phi^{-1}(0.9377) = 1.536\) | B1 | |
| \(\frac{5 - '8.4'}{\sqrt{\frac{490}{n}}} = -1.536\) | M1 | Attempting to standardise – must have correct form |
| \(n = \left(\frac{1.536}{3.4}\right)^2 \times 490\) \((= 100.0048)\) | M1 | Attempting numerical expression for \(n\) or \(\sqrt{n}\) (must have used a 'z' value) may be implied by answer |
| \(n = 100\) | A1 | No errors seen. Must be whole number |
## Question 4(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\bar{x} = \frac{420}{50} = 8.4$ | B1 | |
| $s^2 = \frac{50}{49}\left(\frac{27530}{50} - \left(\frac{420}{50}\right)^2\right)$ | M1 | Or $\frac{1}{49}(27530 - (420)^2/50)$ |
| $= 489.8(36...)$ | A1 | Must see $\geqslant 4$ sf |
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## Question 4(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\Phi^{-1}(0.9377) = 1.536$ | B1 | |
| $\frac{5 - '8.4'}{\sqrt{\frac{490}{n}}} = -1.536$ | M1 | Attempting to standardise – must have correct form |
| $n = \left(\frac{1.536}{3.4}\right)^2 \times 490$ $(= 100.0048)$ | M1 | Attempting numerical expression for $n$ or $\sqrt{n}$ (must have used a 'z' value) may be implied by answer |
| $n = 100$ | A1 | No errors seen. Must be whole number |
---4 The lifetimes, $X$ hours, of a random sample of 50 batteries of a certain kind were found. The results are summarised by $\Sigma x = 420$ and $\Sigma x ^ { 2 } = 27530$.\\
(i) Calculate an unbiased estimate of the population mean of $X$ and show that an unbiased estimate of the population variance is 490 , correct to 3 significant figures.\\
(ii) The lifetimes of a further large sample of $n$ batteries of this kind were noted, and the sample mean, $\bar { X }$, was found. Use your estimates from part (i) to find the value of $n$ such that $\mathrm { P } ( \bar { X } > 5 ) = 0.9377$.\\[0pt]
[4]\\
\hfill \mbox{\textit{CAIE S2 2019 Q4 [7]}}