| Exam Board | OCR |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2015 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Bivariate data |
| Type | Calculate r from summary statistics |
| Difficulty | Moderate -0.8 This is a straightforward application of the PMCC formula using given summary statistics. Part (i) is pure substitution into a standard formula, part (ii) requires basic interpretation of correlation strength, and part (iii) asks for standard reliability concerns (extrapolation, weak correlation). All parts are routine recall and application with no problem-solving or novel insight required. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.09c Calculate regression line |
| Club | Manchester United | Manchester City | Chelsea | Arsenal | Tottenham | Liverpool |
| \(x\) | 5.6 | 7.4 | 6.5 | 4.1 | 3.6 | 6.5 |
| \(y\) | 80 | 71 | 71 | 68 | 62 | 58 |
| Answer | Marks | Guidance |
|---|---|---|
| \(S_{xx} = 200.39 - \frac{33.7^2}{6}\) | M1 | Correct sub in a correct \(S\) formula or correct value of one \(S\) seen; or 11.108 or 11.1 or \(\frac{1333}{120}\) |
| \(S_{yy} = 28314 - \frac{410^2}{6}\) | or 297.333 or 297 or \(\frac{892}{3}\) | |
| \(S_{xy} = 2313.9 - \frac{33.7 \times 410}{6}\) | M1 | Correct sub in 3 correct \(S\) formulae and a correct \(r\) formula; or 11.067 or 11.1 or \(\frac{166}{15}\) |
| \(r = \frac{11.067}{\sqrt{11.108 \times 297.333}}\) | M1 | |
| \(= 0.193\) (3 sf) | A1 | No working: 0.193 M1M1A1; Ignore comment about \(0 < r < 0.2\) |
| Answer | Marks | Guidance |
|---|---|---|
| No/little/poor/weak relationship/corr'n/link between (top) salaries and no. of points | B1 | Allow without "For these 6 clubs" & "top"; or "no strong corr'n between etc" in context; Ignore all else including "positive"; NOT if use "goals" instead of "points" |
| Answer | Marks | Guidance |
|---|---|---|
| Extrapolation | B1 | Outside range of values. Salary is less than the others. |
| Corr'n poor/weak or no rel'nship/link or Points not close to line | B1 | \(r\) small or \(r\) close to 0 or \(r\) not close to 1; or Results do not correlate well; NOT "Corr'n does not imply causation"; NOT "Could be other factors"; NOT if use "goals" instead of "points" |
| Small sample or only (top) 6 clubs | B1 | Any two; allow without context |
# Question 1:
## Part (i)
$S_{xx} = 200.39 - \frac{33.7^2}{6}$ | M1 | Correct sub in a correct $S$ formula or correct value of one $S$ seen; or 11.108 or 11.1 or $\frac{1333}{120}$
$S_{yy} = 28314 - \frac{410^2}{6}$ | | or 297.333 or 297 or $\frac{892}{3}$
$S_{xy} = 2313.9 - \frac{33.7 \times 410}{6}$ | M1 | Correct sub in 3 correct $S$ formulae and a correct $r$ formula; or 11.067 or 11.1 or $\frac{166}{15}$
$r = \frac{11.067}{\sqrt{11.108 \times 297.333}}$ | M1 |
$= 0.193$ (3 sf) | A1 | No working: 0.193 M1M1A1; Ignore comment about $0 < r < 0.2$
**[3]**
## Part (ii)
No/little/poor/weak relationship/corr'n/link between (top) salaries and no. of points | B1 | Allow without "For these 6 clubs" & "top"; or "no strong corr'n between etc" in context; Ignore all else including "positive"; NOT if use "goals" instead of "points"
**[1]**
## Part (iii)
Extrapolation | B1 | Outside range of values. Salary is less than the others.
Corr'n poor/weak or no rel'nship/link or Points not close to line | B1 | $r$ small or $r$ close to 0 or $r$ not close to 1; or Results do not correlate well; NOT "Corr'n does not imply causation"; NOT "Could be other factors"; NOT if use "goals" instead of "points"
Small sample or only (top) 6 clubs | B1 | Any two; allow without context
**[2]**
---1 For the top 6 clubs in the 2010/11 season of the English Premier League, the table shows the annual salary, $\pounds x$ million, of the highest paid player and the number of points scored, $y$.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
Club & Manchester United & Manchester City & Chelsea & Arsenal & Tottenham & Liverpool \\
\hline
$x$ & 5.6 & 7.4 & 6.5 & 4.1 & 3.6 & 6.5 \\
\hline
$y$ & 80 & 71 & 71 & 68 & 62 & 58 \\
\hline
\end{tabular}
\end{center}
$$n = 6 \quad \sum x = 33.7 \quad \sum x ^ { 2 } = 200.39 \quad \sum y = 410 \quad \sum y ^ { 2 } = 28314 \quad \sum x y = 2313.9$$
(i) Use a suitable formula to calculate the product moment correlation coefficient, $r$, between $x$ and $y$, showing that $0 < r < 0.2$.\\
(ii) State what this value of $r$ shows in this context.\\
(iii) A fan suggests that the data should be used to draw a regression line in order to estimate the number of points that would be scored by another Premier League club, whose highest paid player's salary is $\pounds 1.7$ million. Give two reasons why such an estimate would be unlikely to be reliable.
\hfill \mbox{\textit{OCR S1 2015 Q1 [6]}}