| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2015 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Forward transformation: find new statistics |
| Difficulty | Moderate -0.8 This is a straightforward application of standard formulas for mean and variance, followed by routine transformation rules (mean scales linearly, variance scales by the square of the coefficient). Both parts require only direct substitution into well-known formulas with no problem-solving or conceptual insight needed. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left(\frac{1508}{50}\right) = 30.16\ (30.2)\) | B1 | Allow any form |
| \(\frac{50}{49}\left(\frac{51825}{50}-(30.16^2)\right)\) | M1 | (129.46367) |
| \(= 129\) (3 sf) or \(130\) | A1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((1.5 \times\ {}^{\prime}30.16{}^{\prime} + 10) = 55.24\) | B1ft | ft their 30.16 |
| \((1.5^2 \times\ {}^{\prime}129\ldots{}^{\prime})\) | M1 | \(1.5^2 \times \text{their}(129)\) with nothing added at any stage |
| \(= 291\) (3 sf) | A1ft [3] | Allow 290 |
## Question 4:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left(\frac{1508}{50}\right) = 30.16\ (30.2)$ | B1 | Allow any form |
| $\frac{50}{49}\left(\frac{51825}{50}-(30.16^2)\right)$ | M1 | (129.46367) |
| $= 129$ (3 sf) or $130$ | A1 [3] | |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(1.5 \times\ {}^{\prime}30.16{}^{\prime} + 10) = 55.24$ | B1ft | ft their 30.16 |
| $(1.5^2 \times\ {}^{\prime}129\ldots{}^{\prime})$ | M1 | $1.5^2 \times \text{their}(129)$ with nothing added at any stage |
| $= 291$ (3 sf) | A1ft [3] | Allow 290 |
---4 The marks, $x$, of a random sample of 50 students in a test were summarised as follows.
$$n = 50 \quad \Sigma x = 1508 \quad \Sigma x ^ { 2 } = 51825$$
(i) Calculate unbiased estimates of the population mean and variance.\\
(ii) Each student's mark is scaled using the formula $y = 1.5 x + 10$. Find estimates of the population mean and variance of the scaled marks, $y$.
\hfill \mbox{\textit{CAIE S2 2015 Q4 [6]}}