| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2013 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Draw cumulative frequency graph from cumulative frequency table |
| Difficulty | Easy -1.8 This is a routine statistics question requiring straightforward plotting of given cumulative frequency data and reading values from the graph. Part (i) is pure graph plotting with no calculation, part (ii) is simple graph reading (finding the weight where cumulative frequency = 80), and part (iii) uses standard formulas with mid-interval values. All techniques are direct textbook exercises with no problem-solving or insight required. |
| Spec | 2.02a Interpret single variable data: tables and diagrams2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
| Weight \(( x\) kilograms \()\) | \(x < 40\) | \(x < 50\) | \(x < 60\) | \(x < 65\) | \(x < 70\) | \(x < 90\) |
| Cumulative frequency | 0 | 12 | 34 | 64 | 92 | 144 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) (40, 0), (50, 12) etc. up to (90, 144) cf points | B1 | Axes, (cf) and labels (kg), uniform scales from at least 0–140 and 40.5–69.5 either way round |
| B1 [2] | All points correct, sensible scale (not 12), polygon or smooth curve | |
| (ii) 80 weigh less than 67.2 kg | M1 | Subt 64 from 144 |
| \(c = 67.2\) | A1 ft [2] | Accept anything between 67 and 68 ft from incorrect graph |
| (iii) freqs 12, 22, 30, 28, 52 | M1 | frequencies attempt not cf |
| A1 | Correct freqs | |
| mean wt \(= (45 \times 12 + 55 \times 22 + 62.5 \times 30 + 67.5 \times 28 + 80 \times 52) / 144 = 9675 / 144 = 67.2\) kg | M1 | Using mid points attempt, i.e. 44.5, 45, 45.5, in correct mean formula, unsimplified, no cfs, condone 1 error. |
| A1 | Correct mean | |
| Var \((45^2 \times 12 + 55^2 \times 22 + 62.5^2 \times 30 + 67.5^2 \times 28 + 80^2 \times 52) / 144 - (9675/144)^2 = 127.59\) | M1 | Substituting their mid-pts squared (may be class widths, lower or upper bound) in correct var formula even with cfs with their mean² |
| sd \(= 11.3\), allow 11.2 | A1 [6] | Correct answer |
**(i)** (40, 0), (50, 12) etc. up to (90, 144) cf points | B1 | Axes, (cf) and labels (kg), uniform scales from at least 0–140 and 40.5–69.5 either way round
| B1 [2] | All points correct, sensible scale (not 12), polygon or smooth curve
**(ii)** 80 weigh less than 67.2 kg | M1 | Subt 64 from 144
$c = 67.2$ | A1 ft [2] | Accept anything between 67 and 68 ft from incorrect graph
**(iii)** freqs 12, 22, 30, 28, 52 | M1 | frequencies attempt not cf
| A1 | Correct freqs
mean wt $= (45 \times 12 + 55 \times 22 + 62.5 \times 30 + 67.5 \times 28 + 80 \times 52) / 144 = 9675 / 144 = 67.2$ kg | M1 | Using mid points attempt, i.e. 44.5, 45, 45.5, in correct mean formula, unsimplified, no cfs, condone 1 error.
| A1 | Correct mean
Var $(45^2 \times 12 + 55^2 \times 22 + 62.5^2 \times 30 + 67.5^2 \times 28 + 80^2 \times 52) / 144 - (9675/144)^2 = 127.59$ | M1 | Substituting their mid-pts squared (may be class widths, lower or upper bound) in correct var formula even with cfs with their mean²
sd $= 11.3$, allow 11.2 | A1 [6] | Correct answer6 The weights, $x$ kilograms, of 144 people were recorded. The results are summarised in the cumulative frequency table below.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
Weight $( x$ kilograms $)$ & $x < 40$ & $x < 50$ & $x < 60$ & $x < 65$ & $x < 70$ & $x < 90$ \\
\hline
Cumulative frequency & 0 & 12 & 34 & 64 & 92 & 144 \\
\hline
\end{tabular}
\end{center}
(i) On graph paper, draw a cumulative frequency graph to represent these results.\\
(ii) 64 people weigh more than $c \mathrm {~kg}$. Use your graph to find the value of $c$.\\
(iii) Calculate estimates of the mean and standard deviation of the weights.
\hfill \mbox{\textit{CAIE S1 2013 Q6 [10]}}