| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2012 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Data representation |
| Type | Calculate frequency density from frequency |
| Difficulty | Moderate -0.8 This is a straightforward histogram question testing basic understanding of frequency density (frequency รท class width) and cumulative frequency. Part (i) requires simple division, part (ii) is direct recall of cumulative frequency plotting conventions, and part (iii) uses standard formulas with grouped data. All techniques are routine S1 content with no problem-solving or novel insight required. |
| Spec | 2.02b Histogram: area represents frequency2.02g Calculate mean and standard deviation |
| Hours of sunshine | 0 | \(1 - 3\) | \(4 - 6\) | \(7 - 9\) | \(10 - 15\) |
| Number of days | 0 | 6 | 9 | 4 | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((\frac{6}{3} =)\ 2\) | B1 [1] | \((\frac{6}{9} \times 3 =)\ 2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{2}{6} \times 2\) | M1 | Allow \(\frac{2}{5} \times 2\) or ans \(0.8\) for M1 |
| \(= \frac{2}{3}\) oe or \(0.667\) or \(0.67\) or \(0.7\) | A1[2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((3.5,\ 6)\) | B1 | |
| \((0.5,\ 0)\) or \((6.5,\ 15)\) | B1 [2] | Ignore incorrect |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{\Sigma xf}{21}\) | M1 | Allow \(x\) within classes, incl end pts |
| \(= 5.43\) (3 sf) or \(\frac{114}{21}\) or \(\frac{38}{7}\) oe | A1 | then \(\div 5\): M0A0 |
| \(\frac{\Sigma x^2 f}{21}\) or \(\frac{817.5}{21}\) or \(38.9\ldots\) | M1 | Allow \(x\) within class, incl end pt \(\div 5\): M0 |
| \(- \text{"5.43"}^2\) or \(= 9.46\) or \(9.4592\ldots\) | M1 | dep \(+\)ve result; done before \(\sqrt{}\); not \(-(\bar{x}^2 \div \ldots)\) |
| \((\sqrt{9.4592\ldots}) = 3.08\) (3 sf) | A1 [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Actual values or exact hours unknown oe; Don't have raw data oe; or measured to nearest hour oe | B1 [1] | or Data given in classes or grouped oe; or Data evenly distributed in classes oe |
## Question 5:
### Part (i)(a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(\frac{6}{3} =)\ 2$ | B1 [1] | $(\frac{6}{9} \times 3 =)\ 2$ | |
### Part (i)(b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{2}{6} \times 2$ | M1 | Allow $\frac{2}{5} \times 2$ or ans $0.8$ for M1 | Can be implied, eg $\frac{1}{3} = 0.3$, ans $0.6$: M1A0 |
| $= \frac{2}{3}$ oe or $0.667$ or $0.67$ or $0.7$ | A1[2] | | Allow $0.66$ or $0.666$ |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(3.5,\ 6)$ | B1 | | $(6,\ 3.5)$ AND $(15,\ 6.5)$: B1 |
| $(0.5,\ 0)$ or $(6.5,\ 15)$ | B1 [2] | Ignore incorrect | |
### Part (iii)(a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{\Sigma xf}{21}$ | M1 | Allow $x$ within classes, incl end pts | $\geq 2$ non-zero terms correct ft their $x$ |
| $= 5.43$ (3 sf) or $\frac{114}{21}$ or $\frac{38}{7}$ oe | A1 | then $\div 5$: M0A0 | |
| $\frac{\Sigma x^2 f}{21}$ or $\frac{817.5}{21}$ or $38.9\ldots$ | M1 | Allow $x$ within class, incl end pt $\div 5$: M0 | $\geq 2$ non-zero terms correct ft their $x$; Calc 4 values of $(x-\bar{x})^2$ or $(x-\bar{x})^2 f$: or $(11.8, 0.184, 6.61, 50)$ or $(70.5, 1.65, 26.4, 100)$ or $199$ M1; $\frac{\Sigma(x-\bar{x})^2 f}{21}$ fully correct method M1 |
| $- \text{"5.43"}^2$ or $= 9.46$ or $9.4592\ldots$ | M1 | dep $+$ve result; done before $\sqrt{}$; not $-(\bar{x}^2 \div \ldots)$ | |
| $(\sqrt{9.4592\ldots}) = 3.08$ (3 sf) | A1 [5] | | |
### Part (iii)(b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Actual values or exact hours unknown oe; Don't have raw data oe; or measured to nearest hour oe | B1 [1] | or Data given in classes or grouped oe; or Data evenly distributed in classes oe | Mid-points or medians or averages of class boundaries used oe |5 At a certain resort the number of hours of sunshine, measured to the nearest hour, was recorded on each of 21 days. The results are summarised in the table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
Hours of sunshine & 0 & $1 - 3$ & $4 - 6$ & $7 - 9$ & $10 - 15$ \\
\hline
Number of days & 0 & 6 & 9 & 4 & 2 \\
\hline
\end{tabular}
\end{center}
The diagram shows part of a histogram to illustrate the data. The scale on the frequency density axis is 2 cm to 1 unit.\\
\includegraphics[max width=\textwidth, alt={}, center]{56ca7462-d061-48d3-bc5f-274d925e4e34-3_944_1778_699_148}
\begin{enumerate}[label=(\roman*)]
\item (a) Calculate the frequency density of the $1 - 3$ class.\\
(b) Fred wishes to draw the block for the 10 - 15 class on the same diagram. Calculate the height, in centimetres, of this block.
\item A cumulative frequency graph is to be drawn. Write down the coordinates of the first two points that should be plotted. You are not asked to draw the graph.
\item (a) Calculate estimates of the mean and standard deviation of the number of hours of sunshine.\\
(b) Explain why your answers are only estimates.
\end{enumerate}
\hfill \mbox{\textit{OCR S1 2012 Q5 [11]}}