| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2005 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Calculate mean from coded sums |
| Difficulty | Moderate -0.8 This is a straightforward grouped data statistics problem requiring standard formulas for mean and standard deviation. Part (i) involves setting up a simple linear equation using the mean formula (one unknown, basic algebra), and part (ii) is direct application of the standard deviation formula with given class midpoints. Both parts are routine calculations with no conceptual challenges beyond recall of formulas. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation |
| Frequency | ||
| \(0 \leqslant t < 10\) | 2 | ||
| \(10 \leqslant t < 20\) | \(f\) | ||
| \(20 \leqslant t < 40\) | 11 | ||
| \(40 \leqslant t < 80\) | 4 |
total \(= 26\) AG
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| For attempt at LHS, accept end points or cl width | M1 | |
| For attempt at RHS, must have \(17 + f\) | M1 | |
| For correct \(f\) | A1 | |
| For correct answer given, ft if previous answer rounds to 9 | A1 [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| For method including sq rt and mean squared (can be implied if using calculator, must be \(x^2f\) on mid-points) or \(\sum \frac{f(x - \bar{x})^2}{26}\) | M1 | |
| For correct answer | A1 [2] |
(i) $5 \times 2 + 15f + 30 \times 11 + 60 \times 4 = 27.5(17 + f)$
$f = 9$
total $= 26$ AG
| Answer | Marks | Guidance |
|--------|-------|----------|
| For attempt at LHS, accept end points or cl width | M1 | |
| For attempt at RHS, must have $17 + f$ | M1 | |
| For correct $f$ | A1 | |
| For correct answer given, ft if previous answer rounds to 9 | A1 [4] | |
(ii) $\sigma = 16.1$
| Answer | Marks | Guidance |
|--------|-------|----------|
| For method including sq rt and mean squared (can be implied if using calculator, must be $x^2f$ on mid-points) or $\sum \frac{f(x - \bar{x})^2}{26}$ | M1 | |
| For correct answer | A1 [2] | |
---2 The following table shows the results of a survey to find the average daily time, in minutes, that a group of schoolchildren spent in internet chat rooms.
\begin{center}
\begin{tabular}{ | c | c | }
\hline
\begin{tabular}{ c }
Time per day \\
$( t$ minutes $)$ \\
\end{tabular} & Frequency \\
\hline
$0 \leqslant t < 10$ & 2 \\
\hline
$10 \leqslant t < 20$ & $f$ \\
\hline
$20 \leqslant t < 40$ & 11 \\
\hline
$40 \leqslant t < 80$ & 4 \\
\hline
\end{tabular}
\end{center}
The mean time was calculated to be 27.5 minutes.\\
(i) Form an equation involving $f$ and hence show that the total number of children in the survey was 26 .\\
(ii) Find the standard deviation of these times.
\hfill \mbox{\textit{CAIE S1 2005 Q2 [6]}}