| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2018 |
| Session | November |
| Marks | 10 |
| 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 straightforward graph-plotting exercise requiring only the mechanical skill of plotting given coordinate pairs (upper class boundaries against cumulative frequencies) and joining them with a smooth curve. It involves no calculation, problem-solving, or conceptual understanding beyond basic data representation conventions. |
| Spec | 2.02a Interpret single variable data: tables and diagrams |
| Rainfall, \(x \mathrm {~mm}\) | \(x \leqslant 20\) | \(x \leqslant 30\) | \(x \leqslant 40\) | \(x \leqslant 50\) | \(x \leqslant 70\) | \(x \leqslant 100\) |
| Cumulative frequency | 52 | 94 | 142 | 172 | 222 | 250 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Cumulative frequency graph with appropriate linear scales starting at \((0,0)\), axes labelled cf and Rainfall, mm | B1 | Appropriate linear scales starting at \((0,0)\), axes labelled cf and Rainfall, mm |
| Correct graph, points plotted at ucb | B1 | Correct graph, points plotted at ucb, allow straight lines or curve |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Read off from increasing graph at \(cf = 150\) | M1 | Read off from increasing graph at \(cf = 150\) |
| \(42\) | A1 | Correct answer (\(41 \leq r \leq 43\)) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Frequencies: \(52, 42, 48, 30, 50, 28\) | B1 | Correct frequencies |
| Mean age \(= (10\times52 + 25\times42 + 35\times48 + 45\times30 + 60\times50 + 85\times28)/250\) | B1 | Correct midpoints (allow one error) |
| \(= 9980/250\) | M1 | Using \(\Sigma fx/250\) with mid-points attempt, not cf, cw, lb, ub |
| \(= 39.9(2)\) oe | A1 | Correct answer |
| Variance \(= (10^2\times52 + 25^2\times42 + 35^2\times48 + 45^2\times30 + 60^2\times50 + 85^2\times28)/250 - \text{mean}^2 = 539.59\) | M1 | Attempt at variance using their midpoints and their mean |
| \(\sigma = 23.2\) | A1 | Correct answer for sd |
## Question 6(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Cumulative frequency graph with appropriate linear scales starting at $(0,0)$, axes labelled cf and Rainfall, mm | B1 | Appropriate linear scales starting at $(0,0)$, axes labelled cf and Rainfall, mm |
| Correct graph, points plotted at ucb | B1 | Correct graph, points plotted at ucb, allow straight lines or curve |
## Question 6(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Read off from increasing graph at $cf = 150$ | M1 | Read off from increasing graph at $cf = 150$ |
| $42$ | A1 | Correct answer ($41 \leq r \leq 43$) |
## Question 6(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Frequencies: $52, 42, 48, 30, 50, 28$ | B1 | Correct frequencies |
| Mean age $= (10\times52 + 25\times42 + 35\times48 + 45\times30 + 60\times50 + 85\times28)/250$ | B1 | Correct midpoints (allow one error) |
| $= 9980/250$ | M1 | Using $\Sigma fx/250$ with mid-points attempt, not cf, cw, lb, ub |
| $= 39.9(2)$ oe | A1 | Correct answer |
| Variance $= (10^2\times52 + 25^2\times42 + 35^2\times48 + 45^2\times30 + 60^2\times50 + 85^2\times28)/250 - \text{mean}^2 = 539.59$ | M1 | Attempt at variance using their midpoints and their mean |
| $\sigma = 23.2$ | A1 | Correct answer for sd |6 The daily rainfall, $x \mathrm {~mm}$, in a certain village is recorded on 250 consecutive days. The results are summarised in the following cumulative frequency table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
Rainfall, $x \mathrm {~mm}$ & $x \leqslant 20$ & $x \leqslant 30$ & $x \leqslant 40$ & $x \leqslant 50$ & $x \leqslant 70$ & $x \leqslant 100$ \\
\hline
Cumulative frequency & 52 & 94 & 142 & 172 & 222 & 250 \\
\hline
\end{tabular}
\end{center}
(i) On the grid, draw a cumulative frequency graph to illustrate the data.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
& & & & & & & & & & & & & & \\
\hline
\end{tabular}
\end{center}
(ii) On 100 of the days, the rainfall was $k \mathrm {~mm}$ or more. Use your graph to estimate the value of $k$.\\
(iii) Calculate estimates of the mean and standard deviation of the daily rainfall in this village.\\
\hfill \mbox{\textit{CAIE S1 2018 Q6 [10]}}