| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2022 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Histogram from continuous grouped data |
| Difficulty | Moderate -0.8 This is a straightforward application of standard statistical formulas for grouped data. Part (a) requires calculating frequency densities and drawing bars (routine skill), while part (b) involves using the given mean to calculate standard deviation from grouped data using the formula—a direct textbook exercise with no conceptual challenges or problem-solving required. |
| Spec | 2.02b Histogram: area represents frequency |
| Time taken \(( t\) minutes \()\) | \(0 \leqslant t < 20\) | \(20 \leqslant t < 40\) | \(40 \leqslant t < 50\) | \(50 \leqslant t < 60\) | \(60 \leqslant t < 100\) |
| Frequency | 32 | 46 | 96 | 52 | 24 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Cw: 20, 20, 10, 10, 40; Fd: 1.6, 2.3, 9.6, 5.2, 0.6; histogram drawn | M1 | At least 4 frequency densities calculated \(\frac{f}{cw}\) e.g. \(\frac{32}{20}\); condone \(\frac{f}{cw \pm 0.5}\) if unsimplified; may be read from graph using *their* scale, no lower than 1 cm = fd 1 |
| All bar heights correct on graph | A1 | All bar heights correct using *their* suitable linear scale with at least 3 values indicated, no lower than 1 cm = fd 2 |
| Bar ends at \([0,]\) 20, 40, 50, 60, 100 (at axis), 5 bars drawn | B1 | \(0 \leqslant \text{time} \leqslant 100\), linear scale with at least 3 values indicated |
| Axes labelled frequency density (fd), time (\(t\)) and minutes (mins, m) or appropriate title | B1 | Axes may be reversed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Midpoints 10, 30, 45, 55, 80 | B1 | At least 4 correct midpoints seen |
| \([\text{Mean} = 43.2 \text{ given}]\) \([\text{Var} =] \frac{32\times10^2+46\times30^2+96\times45^2+52\times55^2+24\times80^2}{250} - 43.2^2\) | M1 | Appropriate variance formula with *their* 5 midpoints (not upper bound, lower bound, class width, frequency density, frequency or cumulative frequency). Condone 1 frequency error. If correct midpoints seen accept \(\left\{\frac{3200+41400+194400+157300+153600}{250} \text{ or } \frac{549900}{250}\right\} - \{43.2^2 \text{ or } 1866.24\}\) |
| \(= \left[\frac{549900}{250} - 43.2^2 = 333.36\right]\); Sd \(= 18.3\) | A1 | www, final answer 18.25814887 to at least 3SF. If M0 earned SC B1 for final answer 18.25814887 to at least 3SF |
## Question 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Cw: 20, 20, 10, 10, 40; Fd: 1.6, 2.3, 9.6, 5.2, 0.6; histogram drawn | M1 | At least 4 frequency densities calculated $\frac{f}{cw}$ e.g. $\frac{32}{20}$; condone $\frac{f}{cw \pm 0.5}$ if unsimplified; may be read from graph using *their* scale, no lower than 1 cm = fd 1 |
| All bar heights correct on graph | A1 | All bar heights correct using *their* suitable linear scale with at least 3 values indicated, no lower than 1 cm = fd 2 |
| Bar ends at $[0,]$ 20, 40, 50, 60, 100 (at axis), 5 bars drawn | B1 | $0 \leqslant \text{time} \leqslant 100$, linear scale with at least 3 values indicated |
| Axes labelled frequency density (fd), time ($t$) and minutes (mins, m) or appropriate title | B1 | Axes may be reversed |
---
## Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Midpoints 10, 30, 45, 55, 80 | B1 | At least 4 correct midpoints seen |
| $[\text{Mean} = 43.2 \text{ given}]$ $[\text{Var} =] \frac{32\times10^2+46\times30^2+96\times45^2+52\times55^2+24\times80^2}{250} - 43.2^2$ | M1 | Appropriate variance formula with *their* 5 midpoints (not upper bound, lower bound, class width, frequency density, frequency or cumulative frequency). Condone 1 frequency error. If correct midpoints seen accept $\left\{\frac{3200+41400+194400+157300+153600}{250} \text{ or } \frac{549900}{250}\right\} - \{43.2^2 \text{ or } 1866.24\}$ |
| $= \left[\frac{549900}{250} - 43.2^2 = 333.36\right]$; Sd $= 18.3$ | A1 | www, final answer 18.25814887 to at least 3SF. If M0 earned **SC B1** for final answer 18.25814887 to at least 3SF |
---4 The times taken, in minutes, to complete a word processing task by 250 employees at a particular company are summarised in the table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
Time taken $( t$ minutes $)$ & $0 \leqslant t < 20$ & $20 \leqslant t < 40$ & $40 \leqslant t < 50$ & $50 \leqslant t < 60$ & $60 \leqslant t < 100$ \\
\hline
Frequency & 32 & 46 & 96 & 52 & 24 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Draw a histogram to represent this information.\\
\includegraphics[max width=\textwidth, alt={}, center]{3e74785d-5981-480c-a0fd-f43d5d227f2d-06_1201_1198_1050_516}
From the data, the estimate of the mean time taken by these 250 employees is 43.2 minutes.
\item Calculate an estimate for the standard deviation of these times.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2022 Q4 [7]}}