| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2021 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Draw histogram then estimate mean/standard deviation |
| Difficulty | Moderate -0.3 This is a standard S1 statistics question requiring histogram construction with unequal class widths (requiring frequency density calculation), mean estimation from grouped data using midpoints, and finding the maximum IQR. While it involves multiple parts and careful handling of unequal intervals, these are all routine A-level statistics techniques with no novel problem-solving required. The IQR part requires some thought about extreme positioning within classes but follows standard procedures. Slightly easier than average due to being procedural rather than requiring statistical insight. |
| Spec | 2.02b Histogram: area represents frequency2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
| Time \(( t\) seconds \()\) | \(0 \leqslant t < 10\) | \(10 \leqslant t < 20\) | \(20 \leqslant t < 40\) | \(40 \leqslant t < 60\) | \(60 \leqslant t < 100\) |
| Number of players | 16 | 54 | 78 | 32 | 20 |
| Answer | Marks | Guidance |
|---|---|---|
| Class width | 10 | 10 |
| Frequency Density | 1.6 | 5.4 |
| At least 4 frequency densities calculated | M1 | Accept unsimplified. May be read from graph using *their* scale, 3SF or correct |
| All heights correct on graph | A1 | |
| Bar ends at \(0, 10, 20, \ldots\) with horizontal linear scale, at least 3 values indicated, \(0 \leq\) horizontal axis \(\leq 100\) | B1 | |
| Axes labelled: Frequency density (fd), time (t) and seconds. Linear vertical scale, at least 3 values indicated \(0 \leq\) vertical axis \(\leq 5.4\) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(= \frac{80 + 810 + 2340 + 1600 + 1600}{200}\) | M1 | Uses at least 4 midpoint attempts (e.g. \(5 \pm 0.5\)). Accept unsimplified expression, denominator either correct or *their* \(\Sigma\)frequencies |
| \(\left[\frac{6430}{200} =\right] 32\frac{3}{20}\) or \(32.15\) | A1 | Accept 32.2 |
| Answer | Marks | Guidance |
|---|---|---|
| A value in correct UQ \((40\)–\(60)\) \(-\) a value in correct LQ \((10\)–\(20)\) | M1 | |
| Greatest possible value is \(60 - 10 = 50\) | A1 | Condone \(49.\dot{9}\) |
## Question 5(a):
| Class width | 10 | 10 | 20 | 20 | 40 |
| Frequency Density | 1.6 | 5.4 | 3.9 | 1.6 | 0.5 |
At least 4 frequency densities calculated | M1 | Accept unsimplified. May be read from graph using *their* scale, 3SF or correct
All heights correct on graph | A1 |
Bar ends at $0, 10, 20, \ldots$ with horizontal linear scale, at least 3 values indicated, $0 \leq$ horizontal axis $\leq 100$ | B1 |
Axes labelled: Frequency density (fd), time (t) and seconds. Linear vertical scale, at least 3 values indicated $0 \leq$ vertical axis $\leq 5.4$ | B1 |
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## Question 5(b):
$\text{Mean} = \left[\frac{16\times5 + 54\times15 + 78\times30 + 32\times50 + 20\times80}{200}\right]$
$= \frac{80 + 810 + 2340 + 1600 + 1600}{200}$ | M1 | Uses at least 4 midpoint attempts (e.g. $5 \pm 0.5$). Accept unsimplified expression, denominator either correct or *their* $\Sigma$frequencies
$\left[\frac{6430}{200} =\right] 32\frac{3}{20}$ or $32.15$ | A1 | Accept 32.2
---
## Question 5(c):
A value in correct UQ $(40$–$60)$ $-$ a value in correct LQ $(10$–$20)$ | M1 |
Greatest possible value is $60 - 10 = 50$ | A1 | Condone $49.\dot{9}$
---5 The times taken by 200 players to solve a computer puzzle are summarised in the following table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
Time $( t$ seconds $)$ & $0 \leqslant t < 10$ & $10 \leqslant t < 20$ & $20 \leqslant t < 40$ & $40 \leqslant t < 60$ & $60 \leqslant t < 100$ \\
\hline
Number of players & 16 & 54 & 78 & 32 & 20 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Draw a histogram to represent this information.\\
\includegraphics[max width=\textwidth, alt={}, center]{1a27e2ca-9be5-48a0-a1aa-01844573f4d4-08_1397_1198_808_516}
\item Calculate an estimate of the mean time taken by these 200 players.
\item Find the greatest possible value of the interquartile range of these times.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2021 Q5 [8]}}