| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2022 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Cumulative frequency graph construction then interpretation |
| Difficulty | Moderate -0.8 This is a straightforward statistics question requiring standard techniques: plotting given cumulative frequency points, reading a percentile from the graph, and calculating standard deviation from grouped data. All steps are routine S1 procedures with no problem-solving or novel insight required, making it easier than average A-level maths questions. |
| Spec | 2.02a Interpret single variable data: tables and diagrams |
| Time taken \(( t\) minutes \()\) | \(t \leqslant 20\) | \(t \leqslant 30\) | \(t \leqslant 35\) | \(t \leqslant 40\) | \(t \leqslant 50\) | \(t \leqslant 60\) |
| Cumulative frequency | 32 | 66 | 112 | 178 | 228 | 250 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Cumulative frequency graph plotted with points at \((20,32)\), \((30,66)\), \((35,112)\), \((40,178)\), \((50,228)\), \((60,250)\) | M1 | At least 3 points plotted accurately at class upper end points. Linear cf scale \(0 \leqslant \text{cf} \leqslant 250\) and linear time scale \(0 \leqslant \text{time} \leqslant 60\) with at least 3 values identified on each. |
| All points plotted correct, curve drawn within tolerance, joined to \((0,0)\) | A1 | Axes labelled cumulative frequency (cf), time (t) and minutes (min or m) – or suitable title. Axes can be the other way round. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Line drawn from 150 on cf axis to meet graph at about \(t = 38\) minutes | B1 FT | Must be an increasing cf graph with correct upper bounds. Use of graph must be seen. Expect an answer in range \(37 \leqslant t \leqslant 39\) for a correct graph |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| [Frequencies] \([32]\ 34\ 46\ 66\ 50\ 22\) | B1 | May be unsimplified and/or in variance calculation. |
| [Midpoints] \(10\ 25\ 32.5\ 37.5\ 45\ 55\) | M1 | At least 5 correct midpoints seen, may be unsimplified. |
| \([\text{Variance}] = \dfrac{32\times10^2+34\times25^2+46\times32.5^2+66\times37.5^2+50\times45^2+22\times55^2}{250} - 34.4^2\) | M1 | Correct unsimplified variance formula with *their* midpoints and *their* frequencies for var or sd. (\(-\text{mean}^2\) included) |
| \(\left[= \dfrac{333650}{250} - 34.4^2 = 151.24\right]\) | ||
| \([\text{Sd} =]\ 12.3\) | A1 | Awrt WWW. SC B1 for 12.3 if second M1 not awarded. |
## Question 3(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Cumulative frequency graph plotted with points at $(20,32)$, $(30,66)$, $(35,112)$, $(40,178)$, $(50,228)$, $(60,250)$ | M1 | At least 3 points plotted accurately at class upper end points. Linear cf scale $0 \leqslant \text{cf} \leqslant 250$ and linear time scale $0 \leqslant \text{time} \leqslant 60$ with at least 3 values identified on each. |
| All points plotted correct, curve drawn within tolerance, joined to $(0,0)$ | A1 | Axes labelled cumulative frequency (cf), time (t) and minutes (min or m) – or suitable title. Axes can be the other way round. |
**Total: 2 marks**
---
## Question 3(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Line drawn from 150 on cf axis to meet graph at about $t = 38$ minutes | B1 FT | Must be an increasing cf graph with correct upper bounds. Use of graph must be seen. Expect an answer in range $37 \leqslant t \leqslant 39$ for a correct graph |
**Total: 1 mark**
---
## Question 3(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| [Frequencies] $[32]\ 34\ 46\ 66\ 50\ 22$ | B1 | May be unsimplified and/or in variance calculation. |
| [Midpoints] $10\ 25\ 32.5\ 37.5\ 45\ 55$ | M1 | At least 5 correct midpoints seen, may be unsimplified. |
| $[\text{Variance}] = \dfrac{32\times10^2+34\times25^2+46\times32.5^2+66\times37.5^2+50\times45^2+22\times55^2}{250} - 34.4^2$ | M1 | Correct unsimplified variance formula with *their* midpoints and *their* frequencies for var or sd. ($-\text{mean}^2$ included) |
| $\left[= \dfrac{333650}{250} - 34.4^2 = 151.24\right]$ | | |
| $[\text{Sd} =]\ 12.3$ | A1 | Awrt WWW. **SC B1** for 12.3 if second M1 not awarded. |
**Total: 4 marks**
---3 The times, $t$ minutes, taken to complete a walking challenge by 250 members of a club are summarised in the table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | }
\hline
Time taken $( t$ minutes $)$ & $t \leqslant 20$ & $t \leqslant 30$ & $t \leqslant 35$ & $t \leqslant 40$ & $t \leqslant 50$ & $t \leqslant 60$ \\
\hline
Cumulative frequency & 32 & 66 & 112 & 178 & 228 & 250 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Draw a cumulative frequency graph to illustrate the data.\\
\includegraphics[max width=\textwidth, alt={}, center]{1eb957f4-5088-4991-aa8a-f895d55d2bcf-04_1395_1298_705_466}
\item Use your graph to estimate the 60th percentile of the data.\\
It is given that an estimate for the mean time taken to complete the challenge by these 250 members is 34.4 minutes.
\item Calculate an estimate for the standard deviation of the times taken to complete the challenge by these 250 members.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2022 Q3 [7]}}