| Exam Board | CAIE |
|---|---|
| Module | S2 (Statistics 2) |
| Year | 2024 |
| 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 -1.2 This is a straightforward question testing basic sampling and summary statistics. Part (a) requires simple reading of random digits (843, 109, 665, 021, 76), part (b) uses standard formulas for sample mean and variance with routine arithmetic, and part (c) asks for a simple interpretation. All techniques are direct recall with no problem-solving or insight required. |
| Spec | 2.01c Sampling techniques: simple random, opportunity, etc5.05b Unbiased estimates: of population mean and variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \([567, 109], 665, 21\) | B2 | B1 for each. Allow 021. If more than 2 answers given, count first two and ISW |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\text{Est}(\mu) = \frac{610}{30}\) or \(\frac{61}{3}\) | B1 | OE or 20.3 |
| \(\text{Est}(\sigma^2) = \frac{30}{29}\!\left(\frac{12405}{30} - \left(\frac{610}{30}\right)^2\right)\) or \(\frac{1}{29}\!\left(12405 - \frac{610^2}{30}\right)\) | M1 | Use of correct formula |
| \(= 0.0575\) (3sf) | A1 | Accept \(\frac{5}{87}\) |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Variance is [unrealistically] small so Henri has [probably] made a mistake/claim is [probably] correct | B1 FT | Need both parts. Need 'small' OE, not just \(< 0.1\). FT *their* \(< 0.1\) variance value (not \(-\)ve), e.g. 0.0556 (if omit \(\frac{30}{29}\)). Accept 's.d. \(= 0.24\) is small, so Henri has probably made a mistake'. Note: 'mean is large/small' scores B0, but 'mean large compared to variance so Henri prob made a mistake' scores B1 |
| Total: 1 |
## Question 2:
**Part (a):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $[567, 109], 665, 21$ | B2 | B1 for each. Allow 021. If more than 2 answers given, count first two and ISW |
| **Total: 2** | | |
**Part (b):**
| Answer | Mark | Guidance |
|--------|------|----------|
| $\text{Est}(\mu) = \frac{610}{30}$ or $\frac{61}{3}$ | B1 | OE or 20.3 |
| $\text{Est}(\sigma^2) = \frac{30}{29}\!\left(\frac{12405}{30} - \left(\frac{610}{30}\right)^2\right)$ or $\frac{1}{29}\!\left(12405 - \frac{610^2}{30}\right)$ | M1 | Use of correct formula |
| $= 0.0575$ (3sf) | A1 | Accept $\frac{5}{87}$ |
| **Total: 3** | | |
**Part (c):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Variance is [unrealistically] small so Henri has [probably] made a mistake/claim is [probably] correct | B1 FT | Need both parts. Need 'small' OE, not just $< 0.1$. FT *their* $< 0.1$ variance value (not $-$ve), e.g. 0.0556 (if omit $\frac{30}{29}$). Accept 's.d. $= 0.24$ is small, so Henri has probably made a mistake'. Note: 'mean is large/small' scores B0, but 'mean large compared to variance so Henri prob made a mistake' scores B1 |
| **Total: 1** | | |
---2 Henri wants to choose a random sample from the 804 students at his college. He numbers the students from 1 to 804 and then uses random numbers generated by his calculator. The first 20 random digits produced by his calculator are as follows.
$$\begin{array} { l l l l l l l l l l l l l l l l l l l l }
5 & 6 & 7 & 1 & 0 & 9 & 8 & 4 & 3 & 1 & 0 & 9 & 6 & 6 & 5 & 0 & 2 & 1 & 7 & 6
\end{array}$$
Henri's first two student numbers are 567 and 109.
\begin{enumerate}[label=(\alph*)]
\item Use Henri's digits to find the numbers of the next two students in the sample.\\
There were 30 students in Henri's sample. He asked each of them how much time, $X$ hours, they spent on social media each week, on average. He summarised the results as follows.
$$n = 30 \quad \Sigma x = 610 \quad \Sigma x ^ { 2 } = 12405$$
\item Use this information to calculate an unbiased estimate of the mean of $X$ and show that an unbiased estimate of the variance of $X$ is less than 0.1 .
\item Henri's friend claims that Henri has probably made a mistake in his calculation of $\Sigma x$ or $\Sigma x ^ { 2 }$.
Use your answer to part (b) to comment on this claim.
\end{enumerate}
\hfill \mbox{\textit{CAIE S2 2024 Q2 [6]}}