| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2009 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Draw cumulative frequency graph from frequency table (equal class widths) |
| Difficulty | Moderate -0.8 This is a routine statistics question requiring basic cumulative frequency calculations (finding missing values by addition/subtraction), drawing a standard cumulative frequency graph, reading the median from the graph, and calculating mean/standard deviation from grouped data using standard formulas. All techniques are straightforward applications of AS-level statistics procedures with no problem-solving insight required. |
| Spec | 2.02a Interpret single variable data: tables and diagrams2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
| Frequency |
| ||||
| \(0 < x \leqslant 10\) | 210 | 210 | ||||
| \(10 < x \leqslant 20\) | 134 | 344 | ||||
| \(20 < x \leqslant 30\) | 78 | 422 | ||||
| \(30 < x \leqslant 40\) | 72 | \(a\) | ||||
| \(40 < x \leqslant 60\) | \(b\) | 540 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(a = 494\) | B1 | |
| \(b = 46\) | B1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Correct linear scale minimum 0 to 540 and 0 to 60 | B1 | |
| Labels (cf or people or number of people) and (time, or minutes) and attempt at cf or cf step polygon | B1 | |
| Attempt to plot points at \((10, 210)\), \((20, 344)\), \((30, 422)\), \((40, 494)\) | M1 | |
| Correct graph through \((0,0)\) and \((60, 540)\) | A1 [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Median is 13.5 to 14.6 min | M1 | Attempt to read from graph at line \(y = 270\) or 270.5 |
| A1 [2] | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{(5 \times 210 + 15 \times 134 + 25 \times 78 + 35 \times 72 + 50 \times 46)}{540}\) | M1 | Using midpoints and frequencies |
| \(= \frac{9830}{540} = 18.2\) min | A1 | Correct mean |
| \((5^2 \times 210 + 15^2 \times 134 + \ldots) - 18.2^2\) | M1 | Attempt at \(\frac{\Sigma x^2 f}{\Sigma f} - \bar{x}^2\) numerically; could use cfs, ucb, but not class widths |
| \(\text{sd} = 14.2\) min | A1 [4] | Correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(18.2 \pm 7.1 = 11.1,\ 25.3\) | M1 | Attempt to read their mean \(\pm \frac{1}{2}\) sd from cf graph |
| \(390 - 225 = 155\) to 170 people | A1 [2] | Correct answer |
## Question 6:
### Part (i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a = 494$ | B1 | |
| $b = 46$ | B1 **[2]** | |
### Part (ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Correct linear scale minimum 0 to 540 and 0 to 60 | B1 | |
| Labels (cf or people or number of people) and (time, or minutes) and attempt at cf or cf step polygon | B1 | |
| Attempt to plot points at $(10, 210)$, $(20, 344)$, $(30, 422)$, $(40, 494)$ | M1 | |
| Correct graph through $(0,0)$ and $(60, 540)$ | A1 **[4]** | |
### Part (iii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Median is 13.5 to 14.6 min | M1 | Attempt to read from graph at line $y = 270$ or 270.5 |
| | A1 **[2]** | Correct answer |
### Part (iv)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{(5 \times 210 + 15 \times 134 + 25 \times 78 + 35 \times 72 + 50 \times 46)}{540}$ | M1 | Using midpoints and frequencies |
| $= \frac{9830}{540} = 18.2$ min | A1 | Correct mean |
| $(5^2 \times 210 + 15^2 \times 134 + \ldots) - 18.2^2$ | M1 | Attempt at $\frac{\Sigma x^2 f}{\Sigma f} - \bar{x}^2$ numerically; could use cfs, ucb, but not class widths |
| $\text{sd} = 14.2$ min | A1 **[4]** | Correct answer |
### Part (v)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $18.2 \pm 7.1 = 11.1,\ 25.3$ | M1 | Attempt to read their mean $\pm \frac{1}{2}$ sd from cf graph |
| $390 - 225 = 155$ to 170 people | A1 **[2]** | Correct answer |6 During January the numbers of people entering a store during the first hour after opening were as follows.
\begin{center}
\begin{tabular}{ | c | c | c | }
\hline
\begin{tabular}{ c }
Time after opening, \\
$x$ minutes \\
\end{tabular} & Frequency & \begin{tabular}{ c }
Cumulative \\
frequency \\
\end{tabular} \\
\hline
$0 < x \leqslant 10$ & 210 & 210 \\
\hline
$10 < x \leqslant 20$ & 134 & 344 \\
\hline
$20 < x \leqslant 30$ & 78 & 422 \\
\hline
$30 < x \leqslant 40$ & 72 & $a$ \\
\hline
$40 < x \leqslant 60$ & $b$ & 540 \\
\hline
\end{tabular}
\end{center}
(i) Find the values of $a$ and $b$.\\
(ii) Draw a cumulative frequency graph to represent this information. Take a scale of 2 cm for 10 minutes on the horizontal axis and 2 cm for 50 people on the vertical axis.\\
(iii) Use your graph to estimate the median time after opening that people entered the store.\\
(iv) Calculate estimates of the mean, $m$ minutes, and standard deviation, $s$ minutes, of the time after opening that people entered the store.\\
(v) Use your graph to estimate the number of people entering the store between ( $m - \frac { 1 } { 2 } s$ ) and $\left( m + \frac { 1 } { 2 } s \right)$ minutes after opening.
\hfill \mbox{\textit{CAIE S1 2009 Q6 [14]}}