| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2010 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Calculate using histogram bar dimensions |
| Difficulty | Moderate -0.8 This is a straightforward histogram question requiring basic understanding of frequency density (frequency/class width) and weighted mean calculation. The first part involves simple algebra to find x, and the second part is a standard grouped data mean calculation. Both are routine S1 techniques with no conceptual challenges. |
| Spec | 2.02a Interpret single variable data: tables and diagrams2.02b Histogram: area represents frequency2.02f Measures of average and spread |
| Weight (grams) | \(1 - 10\) | \(11 - 20\) | \(21 - 25\) | \(26 - 30\) | \(31 - 50\) | \(51 - 70\) |
| Frequency | \(2 x\) | \(4 x\) | \(3 x\) | \(5 x\) | \(4 x\) | \(x\) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(3 = 2x/10 \Rightarrow x = 15\) | M1 | Attempt at using freq density = freq / cw |
| A1 | Correct answer | |
| \(\text{height} = \text{freq} / \text{class width} = x/20 = 0.75 \text{ cm}\) | M1 | Attempt at using fd = freq / cw with different cw from above |
| A1 | Correct answer | |
| [4] | ||
| (ii) \(\text{mean wt} = \frac{(5.5 \times 30 + 15.5 \times 60 + 23 \times 45 + 28 \times 75 + 40.5 \times 60 + 60.5 \times 15)}{285}\) | M1 | Using freqs or frequency ratios and midpoints, attempt not ucb, not cw (can do it without x) |
| \(= 26.6 \text{ grams}\) | M1 | Correct unsimplified answer can have fr ratios |
| A1 | Correct answer | |
| [3] |
**(i)** $3 = 2x/10 \Rightarrow x = 15$ | M1 | Attempt at using freq density = freq / cw
| A1 | Correct answer
$\text{height} = \text{freq} / \text{class width} = x/20 = 0.75 \text{ cm}$ | M1 | Attempt at using fd = freq / cw with different cw from above
| A1 | Correct answer
| [4] |
**(ii)** $\text{mean wt} = \frac{(5.5 \times 30 + 15.5 \times 60 + 23 \times 45 + 28 \times 75 + 40.5 \times 60 + 60.5 \times 15)}{285}$ | M1 | Using freqs or frequency ratios and midpoints, attempt not ucb, not cw (can do it without x)
$= 26.6 \text{ grams}$ | M1 | Correct unsimplified answer can have fr ratios
| A1 | Correct answer
| [3] |4 The weights in grams of a number of stones, measured correct to the nearest gram, are represented in the following table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
Weight (grams) & $1 - 10$ & $11 - 20$ & $21 - 25$ & $26 - 30$ & $31 - 50$ & $51 - 70$ \\
\hline
Frequency & $2 x$ & $4 x$ & $3 x$ & $5 x$ & $4 x$ & $x$ \\
\hline
\end{tabular}
\end{center}
A histogram is drawn with a scale of 1 cm to 1 unit on the vertical axis, which represents frequency density. The $1 - 10$ rectangle has height 3 cm .\\
(i) Calculate the value of $x$ and the height of the 51-70 rectangle.\\
(ii) Calculate an estimate of the mean weight of the stones.
\hfill \mbox{\textit{CAIE S1 2010 Q4 [7]}}