| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2018 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Using random number tables/generators |
| Difficulty | Easy -1.3 This question tests routine procedural skills: converting random decimals to sample numbers (multiply by 265, round up), calculating standard unbiased estimators using given formulas, and stating a basic definition. All parts are direct application of memorized techniques with no problem-solving or conceptual depth required. |
| Spec | 2.01c Sampling techniques: simple random, opportunity, etc5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(213, 165, 73, 196\) — Allow \(073\) | B1 | For 3-digit no. \(< 265\), consisting of three consecutive integers from given digits, backwards or forward. (\(73\) or \(073\) counts as a 3-digit no.) |
| B1 | For another three such. Other answers may be valid. If other method used, method must be clear | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{510}{25} = \frac{102}{5}\) or \(20.4\) | B1 | |
| \(\frac{25}{24}\left[\frac{13225}{25} - \left(\frac{102}{5}\right)^2\right]\) | M1 | \(\frac{1}{24}\left(13225 - \frac{510^2}{25}\right)\) |
| \(118\) (3 sf) or \(\frac{2821}{24}\) | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| (Average) weekly earnings of all students in Amy's year | B1 | Not 'All students in Amy's year' |
| Total: 1 |
## Question 2:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $213, 165, 73, 196$ — Allow $073$ | B1 | For 3-digit no. $< 265$, consisting of three consecutive integers from given digits, backwards or forward. ($73$ or $073$ counts as a 3-digit no.) |
| | B1 | For another three such. Other answers may be valid. If other method used, method must be clear |
| **Total: 2** | | |
## Question 2(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{510}{25} = \frac{102}{5}$ or $20.4$ | B1 | |
| $\frac{25}{24}\left[\frac{13225}{25} - \left(\frac{102}{5}\right)^2\right]$ | M1 | $\frac{1}{24}\left(13225 - \frac{510^2}{25}\right)$ |
| $118$ (3 sf) or $\frac{2821}{24}$ | A1 | |
| **Total: 3** | | |
## Question 2(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| (Average) weekly earnings of all students in Amy's year | B1 | Not 'All students in Amy's year' |
| **Total: 1** | | |2 Amy has to choose a random sample from the 265 students in her year at college. She numbers the students from 1 to 265 and then uses random numbers generated by her calculator. The first two random numbers produced by her calculator are 0.213165448 and 0.073165196 .\\
(i) Use these figures to find the numbers of the first four students in her sample.\\
There were 25 students in Amy's sample. She asked each of them how much money, $\$ x$, they earned in a week, on average. Her results are summarised below.
$$n = 25 \quad \Sigma x = 510 \quad \Sigma x ^ { 2 } = 13225$$
(ii) Find unbiased estimates of the population mean and variance.\\
(iii) Explain briefly what is meant by 'population' in this question.\\
\hfill \mbox{\textit{CAIE S2 2018 Q2 [6]}}