| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2015 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Calculate frequency density from frequency |
| Difficulty | Easy -1.8 This is a straightforward application of the frequency density formula (frequency ÷ class width). Part (i) requires simple arithmetic with the relationship frequency = frequency density × class width, and part (ii) is routine histogram drawing. No problem-solving or conceptual insight needed beyond recalling a basic definition. |
| Spec | 2.02b Histogram: area represents frequency |
| Time \(( t\) minutes \()\) | \(60 - 62\) | \(63 - 64\) | \(65 - 67\) | \(68 - 71\) |
| Frequency (number of days) | 3 | 9 | 6 | \(b\) |
| Frequency density | 1 | \(a\) | 2 | 1.5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(a = 9/\text{cw} = 9/2 = 4.5\) | M1 | Using \(\text{fd} = f/\text{cw}\) |
| \(1.5 = b/4\) so \(b = 6\) | A1 | Correct \(a\) |
| A1 [3] | Correct \(b\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Histogram with correct heights (ft their \(b\)) | B1\(\checkmark\) | Correct heights ft their \(b\) |
| Correct widths: 3, 2, 3, 4 starting either 60 or 59.5 | B1 | Correct widths |
| Labels: fd, time or minutes; squiggle and bars from 59.5 to 71.5 | B1 [3] | Labels fd, time/minutes and squiggle and bars from 59.5 to 71.5 |
## Question 3 (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a = 9/\text{cw} = 9/2 = 4.5$ | M1 | Using $\text{fd} = f/\text{cw}$ |
| $1.5 = b/4$ so $b = 6$ | A1 | Correct $a$ |
| | A1 [3] | Correct $b$ |
## Question 3 (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Histogram with correct heights (ft their $b$) | B1$\checkmark$ | Correct heights ft their $b$ |
| Correct widths: 3, 2, 3, 4 starting either 60 or 59.5 | B1 | Correct widths |
| Labels: fd, time or minutes; squiggle and bars from 59.5 to 71.5 | B1 [3] | Labels fd, time/minutes and squiggle and bars from 59.5 to 71.5 |
---3 Robert has a part-time job delivering newspapers. On a number of days he noted the time, correct to the nearest minute, that it took him to do his job. Robert used his results to draw up the following table; two of the values in the table are denoted by $a$ and $b$.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | }
\hline
Time $( t$ minutes $)$ & $60 - 62$ & $63 - 64$ & $65 - 67$ & $68 - 71$ \\
\hline
Frequency (number of days) & 3 & 9 & 6 & $b$ \\
\hline
Frequency density & 1 & $a$ & 2 & 1.5 \\
\hline
\end{tabular}
\end{center}
(i) Find the values of $a$ and $b$.\\
(ii) On graph paper, draw a histogram to represent Robert's times.
\hfill \mbox{\textit{CAIE S1 2015 Q3 [6]}}