| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2020 |
| Session | November |
| 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 points and reading values from the graph. Part (a) is pure graph plotting with no calculation, part (b) is simple graph reading (finding 76th percentile), and part (c) 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 |
| Time taken \(( t\) minutes \()\) | \(t \leqslant 20\) | \(t \leqslant 30\) | \(t \leqslant 40\) | \(t \leqslant 60\) | \(t \leqslant 100\) |
| Cumulative frequency | 12 | 48 | 106 | 134 | 150 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| At least 4 points plotted at upper end points, with both scales linear and at least 3 values indicated | M1 | At least 4 points plotted at upper end points, with both scales linear with at least 3 values indicated |
| Correct cumulative frequency curve | A1 | All plotted correctly with curve drawn joined to \((0, 0)\), axes labelled cumulative frequency, time, minutes |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(150 \times 0.76 = 114\) | M1 | 114 SOI, may be on graph |
| \(k = 45\) (mins) | A1 FT | Clear indication that *their* graph has been used, tolerance \(\pm 1\)mm |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Frequencies: 12 36 58 28 16 | B1 | Correct frequencies seen |
| \(\text{Mean} = \frac{10\times12 + 25\times36 + 35\times58 + 50\times28 + 80\times16}{150}\) | B1 | At least 4 correct midpoints seen and used |
| \(\frac{120 + 900 + 2030 + 1400 + 1280}{150}\) | M1 | Correct formula with *their* midpoints (not upper boundary, lower boundary, class width or frequency density) |
| \(38.2,\ 38\frac{1}{5}\) | A1 | |
| \(\text{Variance} = \frac{12\times10^2 + 36\times25^2 + 58\times35^2 + 28\times50^2 + 16\times80^2}{150} - \text{mean}^2\) | M1 | Substitute *their* midpoints and frequencies (condone use of cumulative frequency) in correct variance formula, must have \(-\textit{their}\ \text{mean}^2\) |
| \(= \frac{1200 + 22500 + 71050 + 70000 + 102400}{150} - \text{mean}^2\) | ||
| Standard deviation \(= \sqrt{321.76} = 17.9\) | A1 | |
| 6 |
## Question 6(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| At least 4 points plotted at upper end points, with both scales linear and at least 3 values indicated | M1 | At least 4 points plotted at upper end points, with both scales linear with at least 3 values indicated |
| Correct cumulative frequency curve | A1 | All plotted correctly with curve drawn joined to $(0, 0)$, axes labelled cumulative frequency, time, minutes |
| | **2** | |
## Question 6(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $150 \times 0.76 = 114$ | M1 | 114 SOI, may be on graph |
| $k = 45$ (mins) | A1 FT | Clear indication that *their* graph has been used, tolerance $\pm 1$mm |
| | **2** | |
## Question 6(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Frequencies: 12 36 58 28 16 | B1 | Correct frequencies seen |
| $\text{Mean} = \frac{10\times12 + 25\times36 + 35\times58 + 50\times28 + 80\times16}{150}$ | B1 | At least 4 correct midpoints seen and used |
| $\frac{120 + 900 + 2030 + 1400 + 1280}{150}$ | M1 | Correct formula with *their* midpoints (not upper boundary, lower boundary, class width or frequency density) |
| $38.2,\ 38\frac{1}{5}$ | A1 | |
| $\text{Variance} = \frac{12\times10^2 + 36\times25^2 + 58\times35^2 + 28\times50^2 + 16\times80^2}{150} - \text{mean}^2$ | M1 | Substitute *their* midpoints and frequencies (condone use of cumulative frequency) in correct variance formula, must have $-\textit{their}\ \text{mean}^2$ |
| $= \frac{1200 + 22500 + 71050 + 70000 + 102400}{150} - \text{mean}^2$ | | |
| Standard deviation $= \sqrt{321.76} = 17.9$ | A1 | |
| | **6** | |6 The times, $t$ minutes, taken by 150 students to complete a particular challenge are summarised in the following cumulative frequency table.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
Time taken $( t$ minutes $)$ & $t \leqslant 20$ & $t \leqslant 30$ & $t \leqslant 40$ & $t \leqslant 60$ & $t \leqslant 100$ \\
\hline
Cumulative frequency & 12 & 48 & 106 & 134 & 150 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Draw a cumulative frequency graph to illustrate the data.\\
\includegraphics[max width=\textwidth, alt={}, center]{033ceb76-8fd4-4a89-ab05-5e20039d1c8d-08_1689_1195_744_516}
\item $24 \%$ of the students take $k$ minutes or longer to complete the challenge. Use your graph to estimate the value of $k$.
\item Calculate estimates of the mean and the standard deviation of the time taken to complete the challenge.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2020 Q6 [10]}}