| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2019 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Draw cumulative frequency graph from cumulative frequency table |
| Difficulty | Easy -1.8 This is a routine, textbook-standard cumulative frequency question requiring only direct plotting of given points, reading values from the graph, and applying the standard mid-interval formula for mean from grouped data. All techniques are straightforward recall with no problem-solving or conceptual challenge. |
| Spec | 2.02a Interpret single variable data: tables and diagrams2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
| Time \(( t\) minutes \()\) | \(t \leqslant 10\) | \(t \leqslant 20\) | \(t \leqslant 30\) | \(t \leqslant 50\) | \(t \leqslant 70\) | \(t \leqslant 90\) |
| Cumulative frequency | 16 | 50 | 106 | 146 | 176 | 200 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Correct labels and scales | B1 | Axes labelled 'cumulative frequency' (or cf) and 'time (or t) [in] min(utes)', linear scales from 0 to 90 and 0 to 200 with at least 3 values marked on each axis |
| 7 correctly plotted points above upper boundaries joined in a curve or line segments | B1 | \((0,0)\); \((10,16)\); \((20,50)\); \((30,106)\); \((50,146)\); \((70,176)\); \((90,200)\) |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(29\) | B1 | \(28 \leqslant \text{median} \leqslant 30\) |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| 120 seen | M1 | For seeing 120 in a calculation or marked on the graph |
| \(37\) | A1FT | \(36 \leqslant \text{Ans} \leqslant 39\) or FT from *their* graph; SC1 unsupported answer in range |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Frequencies 16 34 56 40 30 24 | B1 | Seen. Allow unsimplified |
| Est. Mean \(= \frac{5\times16+15\times34+25\times56+40\times40+60\times30+80\times24}{200}\) | M1 | At least 4 correct midpoints \((5, 15, 25, 40, 60, 80)\) used in a calculation |
| \(\frac{7310}{200}\) | M1 | Summing products of *their* 6 mid-points (not lower or upper bound or class width) \(\times\) *their* frequencies \(/ 200\) (or *their* \(\Sigma f\)), unsimplified |
| \(36.55\) | A1 | Accept 36.6 |
| 4 |
## Question 5(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct labels and scales | B1 | Axes labelled 'cumulative frequency' (or cf) and 'time (or t) [in] min(utes)', linear scales from 0 to 90 and 0 to 200 with at least 3 values marked on each axis |
| 7 correctly plotted points above upper boundaries joined in a curve or line segments | B1 | $(0,0)$; $(10,16)$; $(20,50)$; $(30,106)$; $(50,146)$; $(70,176)$; $(90,200)$ |
| | **2** | |
---
## Question 5(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $29$ | B1 | $28 \leqslant \text{median} \leqslant 30$ |
| | **1** | |
---
## Question 5(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| 120 seen | M1 | For seeing 120 in a calculation or marked on the graph |
| $37$ | A1FT | $36 \leqslant \text{Ans} \leqslant 39$ or FT from *their* graph; **SC1** unsupported answer in range |
| | **2** | |
---
## Question 5(iv):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Frequencies 16 34 56 40 30 24 | B1 | Seen. Allow unsimplified |
| Est. Mean $= \frac{5\times16+15\times34+25\times56+40\times40+60\times30+80\times24}{200}$ | M1 | At least 4 correct midpoints $(5, 15, 25, 40, 60, 80)$ used in a calculation |
| $\frac{7310}{200}$ | M1 | Summing products of *their* 6 mid-points (not lower or upper bound or class width) $\times$ *their* frequencies $/ 200$ (or *their* $\Sigma f$), unsimplified |
| $36.55$ | A1 | Accept 36.6 |
| | **4** | |
---5 Last Saturday, 200 drivers entering a car park were asked the time, in minutes, that it had taken them to travel from home to the car park. The results are summarised in the following cumulative frequency table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
Time $( t$ minutes $)$ & $t \leqslant 10$ & $t \leqslant 20$ & $t \leqslant 30$ & $t \leqslant 50$ & $t \leqslant 70$ & $t \leqslant 90$ \\
\hline
Cumulative frequency & 16 & 50 & 106 & 146 & 176 & 200 \\
\hline
\end{tabular}
\end{center}
(i) On the grid, draw a cumulative frequency graph to illustrate the data.\\
\includegraphics[max width=\textwidth, alt={}, center]{06f6c8dd-170c-4e94-a960-0c649a7363a1-08_1198_1399_735_415}\\
(ii) Use your graph to estimate the median of the data.\\
(iii) For 80 of the drivers, the time taken was at least $T$ minutes. Use your graph to estimate the value of $T$.\\
(iv) Calculate an estimate of the mean time taken by all 200 drivers to travel to the car park.\\
\hfill \mbox{\textit{CAIE S1 2019 Q5 [9]}}