| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2024 |
| Session | March |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Draw histogram then perform other calculations |
| Difficulty | Moderate -0.8 This is a straightforward S1 statistics question requiring standard histogram construction (calculating frequency densities for unequal class widths), mean estimation from grouped data using midpoints, and identifying the quartile position. All techniques are routine textbook exercises with no problem-solving insight required, making it easier than average A-level questions. |
| Spec | 2.02b Histogram: area represents frequency2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
| \(0 \leqslant t < 20\) | \(20 \leqslant t < 30\) | \(30 \leqslant t < 35\) | \(35 \leqslant t < 40\) | \(40 \leqslant t < 50\) | \(50 \leqslant t < 70\) | ||
| Frequency | 8 | 23 | 35 | 52 | 20 | 12 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Class widths: 20, 10, 5, 5, 10, 20; Frequency densities: 0.4, 2.3, 7.0, 10.4, 2.0, 0.6 | M1 | At least five frequency densities (f/cw), e.g. \(\dfrac{8}{20}, \dfrac{23}{10}, \ldots\) Accept unsimplified |
| All heights correct on graph | A1 | All heights correct on graph (no FT) |
| Bar ends at \([0,]\) 20, 30, 35, 40, 50, 70 with linear scale, at least three values indicated | B1 | Bar ends at \([0,]\) 20, 30, 35, 40, 50, 70 with a linear scale and at least three values indicated 'linearly' |
| Axes labelled frequency density (fd) and time (mins) | B1 | Frequency density scale vertical starts at 0 with a linear scale and at least three values indicated 'linearly'. Axes can be reversed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Midpoints 10, 25, 32.5, 37.5, 45, 60 | B1 | At least five correct mid-points seen or used in formula |
| \(\text{Mean} = \dfrac{10\times8 + 25\times23 + 32.5\times35 + 37.5\times52 + 45\times20 + 60\times12}{150}\) \(\left[= \dfrac{5362.5}{150}\right]\) | M1 | Correct mean formula using their 6 midpoints (must be within class, not upper/lower bound). Condone one error. If correct midpoints seen, accept \(\dfrac{5362.5}{150}\) or \(\dfrac{80+575+1137.5+1950+900+720}{150}\) |
| \(= 35.75,\ 35\dfrac{3}{4}\) | A1 | Accept 35.8, not \(\dfrac{143}{4}\). If A0 scored, SC B1 for \(35.75, 35\dfrac{3}{4}\) only |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(30 \leqslant t < 35\) | B1 | Condone '3rd' interval, \(30 - 35\) |
## Question 3(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Class widths: 20, 10, 5, 5, 10, 20; Frequency densities: 0.4, 2.3, 7.0, 10.4, 2.0, 0.6 | M1 | At least five frequency densities (f/cw), e.g. $\dfrac{8}{20}, \dfrac{23}{10}, \ldots$ Accept unsimplified |
| All heights correct on graph | A1 | All heights correct on graph (no FT) |
| Bar ends at $[0,]$ 20, 30, 35, 40, 50, 70 with linear scale, at least three values indicated | B1 | Bar ends at $[0,]$ 20, 30, 35, 40, 50, 70 with a linear scale and at least three values indicated 'linearly' |
| Axes labelled frequency density (fd) and time (mins) | B1 | Frequency density scale vertical starts at 0 with a linear scale and at least three values indicated 'linearly'. Axes can be reversed |
**Total: 4 marks**
---
## Question 3(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Midpoints 10, 25, 32.5, 37.5, 45, 60 | B1 | At least five correct mid-points seen or used in formula |
| $\text{Mean} = \dfrac{10\times8 + 25\times23 + 32.5\times35 + 37.5\times52 + 45\times20 + 60\times12}{150}$ $\left[= \dfrac{5362.5}{150}\right]$ | M1 | Correct mean formula using their 6 midpoints (must be within class, not upper/lower bound). Condone one error. If correct midpoints seen, accept $\dfrac{5362.5}{150}$ or $\dfrac{80+575+1137.5+1950+900+720}{150}$ |
| $= 35.75,\ 35\dfrac{3}{4}$ | A1 | Accept 35.8, not $\dfrac{143}{4}$. If A0 scored, SC B1 for $35.75, 35\dfrac{3}{4}$ only |
**Total: 3 marks**
---
## Question 3(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $30 \leqslant t < 35$ | B1 | Condone '3rd' interval, $30 - 35$ |
**Total: 1 mark**
---3 The times taken, in minutes, by 150 students to complete a puzzle are summarised in the table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
\begin{tabular}{ c }
Time taken \\
$( t$ minutes $)$ \\
\end{tabular} & $0 \leqslant t < 20$ & $20 \leqslant t < 30$ & $30 \leqslant t < 35$ & $35 \leqslant t < 40$ & $40 \leqslant t < 50$ & $50 \leqslant t < 70$ \\
\hline
Frequency & 8 & 23 & 35 & 52 & 20 & 12 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Draw a histogram to represent this information.\\
\includegraphics[max width=\textwidth, alt={}, center]{d1a3524c-a3b5-45fe-86a7-5cbda087efcd-06_1193_1489_886_328}
\item Calculate an estimate for the mean time for these students to complete the puzzle.
\item In which class interval does the lower quartile of the times lie?
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2024 Q3 [8]}}