| Exam Board | OCR MEI |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2010 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Measures of Location and Spread |
| Type | Forward transformation: find new statistics |
| Difficulty | Moderate -0.8 This is a straightforward application of standard formulas for mean and standard deviation from a frequency table, followed by routine linear transformation rules (multiply by constant). Both parts require only direct recall and calculation with no problem-solving or conceptual insight needed. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation5.02c Linear coding: effects on mean and variance |
| Number of loaves | 0 | 1 | 2 | 3 | 4 | 5 |
| Frequency | 37 | 23 | 11 | 3 | 0 | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Mean \(= \frac{0 \times 37 + 1 \times 23 + 2 \times 11 + 3 \times 3 + 4 \times 0 + 5 \times 1}{75} = \frac{59}{75} = 0.787\) | M1, A1 | |
| \(S_{xx} = 0^2 \times 37 + 1^2 \times 23 + 2^2 \times 11 + 3^2 \times 3 + 4^2 \times 0 + 5^2 \times 1 - \frac{59^2}{75} = 72.59\) | M1 for \(\sum fx^2\) s.o.i., M1 dep for good attempt at \(S_{xx}\) BUT NOTE M1M0 if their \(S_{xx} < 0\) | |
| \(s = \sqrt{\frac{72.59}{74}} = 0.99\) | A1 CAO | 5 marks |
| (ii) New mean \(= 0.787 \times £1.04 = £0.818\) or \(81.8\) pence | B1 If their mean, B1 ft their s | |
| New \(s = 0.99 \times £1.04 = £1.03\) or \(103\) pence | B1 for correct units dep on at least 1 correct (ft) | 3 marks |
**(i)** Mean $= \frac{0 \times 37 + 1 \times 23 + 2 \times 11 + 3 \times 3 + 4 \times 0 + 5 \times 1}{75} = \frac{59}{75} = 0.787$ | M1, A1 |
$S_{xx} = 0^2 \times 37 + 1^2 \times 23 + 2^2 \times 11 + 3^2 \times 3 + 4^2 \times 0 + 5^2 \times 1 - \frac{59^2}{75} = 72.59$ | M1 for $\sum fx^2$ s.o.i., M1 dep for good attempt at $S_{xx}$ BUT NOTE M1M0 if their $S_{xx} < 0$ |
$s = \sqrt{\frac{72.59}{74}} = 0.99$ | A1 CAO | 5 marks
**(ii)** New mean $= 0.787 \times £1.04 = £0.818$ or $81.8$ pence | B1 If their mean, B1 ft their s |
New $s = 0.99 \times £1.04 = £1.03$ or $103$ pence | B1 for correct units dep on at least 1 correct (ft) | 3 marks
**TOTAL: 8 marks**
---5 A retail analyst records the numbers of loaves of bread of a particular type bought by a sample of shoppers in a supermarket.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
Number of loaves & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline
Frequency & 37 & 23 & 11 & 3 & 0 & 1 \\
\hline
\end{tabular}
\end{center}
(i) Calculate the mean and standard deviation of the numbers of loaves bought per person.\\
(ii) Each loaf costs $\pounds 1.04$. Calculate the mean and standard deviation of the amount spent on loaves per person.
\hfill \mbox{\textit{OCR MEI S1 2010 Q5 [8]}}