| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2020 |
| Session | March |
| Marks | 9 |
| 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 S1 question requiring standard cumulative frequency graph construction, reading a percentile from the graph, and calculating variance from grouped data using the given mean. All techniques are routine textbook exercises with no problem-solving insight required, making it easier than average A-level maths. |
| Spec | 2.02a Interpret single variable data: tables and diagrams2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
| Length (cm) | \(0 - 9\) | \(10 - 14\) | \(15 - 19\) | \(20 - 30\) |
| Frequency | 15 | 48 | 66 | 21 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Cumulative frequencies: \(15, 63, 129, 150\) | B1 | Correct cumulative frequencies seen (may be on graph) |
| Axes: \(0 \leqslant\) horizontal \(\leqslant 30\), \(0 \leqslant\) vertical \(\leqslant 150\), labels: length cm, cf | B1 | Correct axis ranges and labels |
| At least 3 points plotted at upper end points | M1 | Allow \(9, 9.5, 10\) with linear horizontal scale |
| Linear vertical scale, all points at correct upper end points (\(9.5\) etc.), curve drawn accurately, joined to \((0,0)\) | A1 | Condone \((-0.5, 0)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(60\%\) of \(150 = 90\) | M1 | \(90\) seen or implied by use on graph |
| Approx. \(16.5\) [cm] | A1FT | FT *their* increasing cumulative frequency graph. Use of graph must be seen. If no clear evidence of use of graph, SCB1FT correct value from *their* graph |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Midpoints: \(4.75, 12, 17, 25\) | M1 | At least 3 correct midpoints used (\(39449.4375\) implies M1) |
| \(\text{Var} = \dfrac{4.75^2 \times 15 + 12^2 \times 48 + 17^2 \times 66 + 25^2 \times 21}{150} - 15.295^2\) | M1 | Using midpoints \(\pm 0.5\) in correct variance formula, including subtraction of *their* \(\mu^2\) |
| \(= 29.1\) | A1 |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Cumulative frequencies: $15, 63, 129, 150$ | B1 | Correct cumulative frequencies seen (may be on graph) |
| Axes: $0 \leqslant$ horizontal $\leqslant 30$, $0 \leqslant$ vertical $\leqslant 150$, labels: length cm, cf | B1 | Correct axis ranges and labels |
| At least 3 points plotted at upper end points | M1 | Allow $9, 9.5, 10$ with linear horizontal scale |
| Linear vertical scale, all points at correct upper end points ($9.5$ etc.), curve drawn accurately, joined to $(0,0)$ | A1 | Condone $(-0.5, 0)$ |
---
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $60\%$ of $150 = 90$ | M1 | $90$ seen or implied by use on graph |
| Approx. $16.5$ [cm] | A1FT | FT *their* increasing cumulative frequency graph. Use of graph must be seen. If no clear evidence of use of graph, **SCB1FT** correct value from *their* graph |
---
## Question 7(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Midpoints: $4.75, 12, 17, 25$ | M1 | At least 3 correct midpoints used ($39449.4375$ implies M1) |
| $\text{Var} = \dfrac{4.75^2 \times 15 + 12^2 \times 48 + 17^2 \times 66 + 25^2 \times 21}{150} - 15.295^2$ | M1 | Using midpoints $\pm 0.5$ in correct variance formula, including subtraction of *their* $\mu^2$ |
| $= 29.1$ | A1 | |7 Helen measures the lengths of 150 fish of a certain species in a large pond. These lengths, correct to the nearest centimetre, are summarised in the following table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | }
\hline
Length (cm) & $0 - 9$ & $10 - 14$ & $15 - 19$ & $20 - 30$ \\
\hline
Frequency & 15 & 48 & 66 & 21 \\
\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]{f7c0e35d-1889-4e5b-b094-f467052a66cf-10_1593_1296_790_466}
\item 40\% of these fish have a length of $d \mathrm {~cm}$ or more. Use your graph to estimate the value of $d$.\\
The mean length of these 150 fish is 15.295 cm .
\item Calculate an estimate for the variance of the lengths of the fish.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2020 Q7 [9]}}