| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2011 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Standard unbiased estimates calculation |
| Difficulty | Easy -1.2 This question tests basic sampling methodology and routine calculation of sample mean and variance using standard formulas. Part (i) requires explaining random number sampling (straightforward procedural knowledge), part (ii) involves direct application of memorized formulas with simple arithmetic on 8 values, and part (iii) asks for a definition. No problem-solving, proof, or conceptual insight required—purely recall and mechanical computation. |
| Spec | 2.01c Sampling techniques: simple random, opportunity, etc5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Number all members | B1 | |
| Explain the selection of 3-digit random numbers | B1 | |
| Omit repeats OR omit nos. over 750 (until have 8 nos.) | B1 | [3] |
| (ii) Est \((\mu) = 20\) | B1 | |
| Est \((\sigma) = \frac{8}{7}\left(\frac{3636}{8} - 20^2\right)\) | M1 | \(1/7 \times (3636 - 160^2/8)\) |
| \(= \frac{436}{7}\) or \(62.3\) (3 s.f.) | A1 | [3] \((7.89...)^2\) M1A1, but \(7.89...\) only M1A0 |
| (iii) Amounts spent last week in café by all club members | B1 | [1] |
**(i)** Number all members | B1 |
Explain the selection of 3-digit random numbers | B1 |
Omit repeats OR omit nos. over 750 (until have 8 nos.) | B1 | [3]
**(ii)** Est $(\mu) = 20$ | B1 |
Est $(\sigma) = \frac{8}{7}\left(\frac{3636}{8} - 20^2\right)$ | M1 | $1/7 \times (3636 - 160^2/8)$
$= \frac{436}{7}$ or $62.3$ (3 s.f.) | A1 | [3] $(7.89...)^2$ M1A1, but $7.89...$ only M1A0
**(iii)** Amounts spent last week in café by all club members | B1 | [1]Jack has to choose a random sample of 8 people from the 750 members of a sports club.
\begin{enumerate}[label=(\roman*)]
\item Explain fully how he can use random numbers to choose the sample. [3]
\end{enumerate}
Jack asks each person in the sample how much they spent last week in the club café. The results, in dollars, were as follows.
15 \quad 25 \quad 30 \quad 8 \quad 12 \quad 18 \quad 27 \quad 25
\begin{enumerate}[label=(\roman*), resume]
\item Find unbiased estimates of the population mean and variance. [3]
\item Explain briefly what is meant by 'population' in this question. [1]
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2011 Q3 [7]}}