| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2014 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Complete frequency table and histogram together |
| Difficulty | Moderate -0.3 This is a straightforward data representation question requiring standard techniques: reading cumulative frequencies (trivial subtraction), drawing a histogram with unequal class widths (routine but requires care with frequency density), and calculating mean/variance from grouped data using standard formulas. All are textbook procedures with no problem-solving or novel insight required, making it slightly easier than average. |
| Spec | 2.02a Interpret single variable data: tables and diagrams2.02b Histogram: area represents frequency2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
| Time (seconds) | \(< 10.0\) | \(< 10.5\) | \(< 11.0\) | \(< 12.0\) | \(< 12.5\) | \(< 13.5\) |
| Cumulative frequency | 0 | 4 | 10 | 40 | 49 | 57 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(6\) | B1 [1] | Must see in (i) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| freqs: \(4\ 6\ 30\ 9\ 8\); fd: \(8\ 12\ 30\ 18\ 8\) | M1 | Attempt at scaled frequency or fd (must be f/cw) at least three f/cw |
| [histogram with correct heights] | A1 | Correct heights seen on graph |
| B1 | Correct-looking widths from 10, 10.5 etc.; no gaps, no extra lines | |
| B1 [4] | Labels and linear axes or squiggle; need time or secs, fd |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(E(X) = (10.25 \times 4 + 10.75 \times 6 + 11.5 \times 30 + 12.25 \times 9 + 13 \times 8)/57\) | M1 | Using mid-point attempt (not end points) with their frequency or cf at least 2 sensible ones |
| \(= 11.7\ (11.662)\) | A1 | Correct mean |
| \(\text{Var}(X) = (10.25^2 \times 4 + 10.75^2 \times 6 + 11.5^2 \times 30 + 12.25^2 \times 9 + 13^2 \times 8)/57 - (11.662\ldots)^2\) | M1 | Numerical attempt at correct variance formula with mean\(^2\) subtracted ft their "midpoints" i.e. ucb, cw, etc. |
| \(= 0.547\) | A1 [4] | Accept answers between \(0.547\) and \(0.610\); condone \(0.6\), \(0.60\) |
## Question 6:
### Part (i):
| Working | Mark | Guidance |
|---------|------|----------|
| $6$ | B1 **[1]** | Must see in (i) |
### Part (ii):
| Working | Mark | Guidance |
|---------|------|----------|
| freqs: $4\ 6\ 30\ 9\ 8$; fd: $8\ 12\ 30\ 18\ 8$ | M1 | Attempt at scaled frequency or fd (must be f/cw) at least three f/cw |
| [histogram with correct heights] | A1 | Correct heights seen on graph |
| | B1 | Correct-looking widths from 10, 10.5 etc.; no gaps, no extra lines |
| | B1 **[4]** | Labels and linear axes or squiggle; need time or secs, fd |
### Part (iii):
| Working | Mark | Guidance |
|---------|------|----------|
| $E(X) = (10.25 \times 4 + 10.75 \times 6 + 11.5 \times 30 + 12.25 \times 9 + 13 \times 8)/57$ | M1 | Using mid-point attempt (not end points) with their frequency or cf at least 2 sensible ones |
| $= 11.7\ (11.662)$ | A1 | Correct mean |
| $\text{Var}(X) = (10.25^2 \times 4 + 10.75^2 \times 6 + 11.5^2 \times 30 + 12.25^2 \times 9 + 13^2 \times 8)/57 - (11.662\ldots)^2$ | M1 | Numerical attempt at correct variance formula with mean$^2$ subtracted ft their "midpoints" i.e. ucb, cw, etc. |
| $= 0.547$ | A1 **[4]** | Accept answers between $0.547$ and $0.610$; condone $0.6$, $0.60$ |
---6 The times taken by 57 athletes to run 100 metres are summarised in the following cumulative frequency table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
Time (seconds) & $< 10.0$ & $< 10.5$ & $< 11.0$ & $< 12.0$ & $< 12.5$ & $< 13.5$ \\
\hline
Cumulative frequency & 0 & 4 & 10 & 40 & 49 & 57 \\
\hline
\end{tabular}
\end{center}
(i) State how many athletes ran 100 metres in a time between 10.5 and 11.0 seconds.\\
(ii) Draw a histogram on graph paper to represent the times taken by these athletes to run 100 metres.\\
(iii) Calculate estimates of the mean and variance of the times taken by these athletes.
\hfill \mbox{\textit{CAIE S1 2014 Q6 [9]}}