| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2017 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Cumulative distribution functions |
| Type | CDF from grouped data |
| Difficulty | Moderate -0.8 This is a straightforward grouped data CDF question requiring plotting cumulative frequencies and reading off a value for interpolation. It involves only standard textbook procedures (calculating cumulative frequencies, plotting points, linear interpolation) with no conceptual challenges or problem-solving insight needed. Easier than average A-level content. |
| Spec | 2.02a Interpret single variable data: tables and diagrams |
| Circumference \(( c \mathrm {~cm} )\) | \(40 < c \leqslant 50\) | \(50 < c \leqslant 80\) | \(80 < c \leqslant 100\) | \(100 < c \leqslant 120\) |
| Frequency | 14 | 48 | 70 | 8 |
| Answer | Marks | Guidance |
|---|---|---|
| Points \((50, 14),\ (80, 62),\ (100, 132),\ (120, 140)\) | B1 | Correct cf values seen listed, in or by table or on graph, 0 not required |
| Axes labelled correctly | B1 | Axes labelled 'cumulative frequency' (or cf) and 'circumference [or cir or c etc.] (in) cm'. Linear scales – c.f. 0–140, circumference 40–120 (ignore \(<\)40 on circ.). At least 3 values stated on each axis, but \((0,0)\) can be implied without stating. |
| All points plotted accurately | B1 | All points plotted accurately |
| Answer | Marks | Guidance |
|---|---|---|
| \(140 - 54 = 86\) | M1 | Finding correct value from graph (checked \(\pm 1\) mm) or linear interpolation. Subtraction from 140 can be implied |
| Percentage \(= 61.4\%\) | A1 | \(60.5\% \leqslant \text{Ans} \leqslant 64.5\%\) |
## Question 2(i):
Points $(50, 14),\ (80, 62),\ (100, 132),\ (120, 140)$ | **B1** | Correct cf values seen listed, in or by table or on graph, 0 not required
Axes labelled correctly | **B1** | Axes labelled 'cumulative frequency' (or cf) and 'circumference [or cir or c etc.] (in) cm'. Linear scales – c.f. 0–140, circumference 40–120 (ignore $<$40 on circ.). At least 3 values stated on each axis, but $(0,0)$ can be implied without stating.
All points plotted accurately | **B1** | All points plotted accurately
**Total: 3 marks**
---
## Question 2(ii):
$140 - 54 = 86$ | **M1** | Finding correct value from graph (checked $\pm 1$ mm) or linear interpolation. Subtraction from 140 can be implied
Percentage $= 61.4\%$ | **A1** | $60.5\% \leqslant \text{Ans} \leqslant 64.5\%$
**Total: 2 marks**
---2 The circumferences, $c \mathrm {~cm}$, of some trees in a wood were measured. The results are summarised in the table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | }
\hline
Circumference $( c \mathrm {~cm} )$ & $40 < c \leqslant 50$ & $50 < c \leqslant 80$ & $80 < c \leqslant 100$ & $100 < c \leqslant 120$ \\
\hline
Frequency & 14 & 48 & 70 & 8 \\
\hline
\end{tabular}
\end{center}
(i) On the grid, draw a cumulative frequency graph to represent the information.\\
\includegraphics[max width=\textwidth, alt={}, center]{9c23b94b-e573-4e13-be90-e63a0daf18e5-03_1401_1404_854_413}\\
(ii) Estimate the percentage of trees which have a circumference larger than 75 cm .\\
\hfill \mbox{\textit{CAIE S1 2017 Q2 [5]}}