| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2014 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Bivariate data |
| Type | Calculate r from summary statistics |
| Difficulty | Moderate -0.3 This is a straightforward calculation of the product moment correlation coefficient from given data values, requiring only the standard formula and careful arithmetic. The conceptual part (ii) about the invariance of correlation under linear transformation is a standard result that students should know. While it requires accurate calculation with 10 data points, it's a routine S1 exercise with no problem-solving or novel insight required. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08b Linear coding: effect on pmcc |
| \cline { 2 - 11 } \multicolumn{1}{c|}{} | Rider | ||||||||||
| \cline { 2 - 11 } \multicolumn{1}{c|}{} | \(\mathbf { A }\) | \(\mathbf { B }\) | \(\mathbf { C }\) | \(\mathbf { D }\) | \(\mathbf { E }\) | \(\mathbf { F }\) | \(\mathbf { G }\) | \(\mathbf { H }\) | \(\mathbf { I }\) | \(\mathbf { J }\) | |
| \(\boldsymbol { u }\) | 7.88 | 13.02 | 4.29 | 2.88 | 6.26 | 7.03 | 3.60 | 11.78 | 13.15 | 11.69 | |
| \(\boldsymbol { v }\) | 6.63 | 10.16 | 3.63 | 0.47 | 5.70 | 8.01 | 3.30 | 7.31 | 13.08 | 11.82 | |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\sum u = 81.58\), \(\sum v = 70.11\) | B1 | May be implied |
| \(\sum u^2 = 832.0708\), \(\sum v^2 = 649.0983\), \(\sum uv = 733.3155\) | M1 | Correct method for \(S_{uu}\), \(S_{vv}\), \(S_{uv}\) |
| \(S_{uu} = 832.0708 - \frac{81.58^2}{10} = 166.7456\) | ||
| \(S_{vv} = 649.0983 - \frac{70.11^2}{10} = 157.7089\) | ||
| \(S_{uv} = 733.3155 - \frac{81.58 \times 70.11}{10} = 161.7032\) | ||
| \(r_{uv} = \frac{161.7032}{\sqrt{166.7456 \times 157.7089}}\) | M1 | Correct formula |
| \(r_{uv} = 0.9965\) (awrt \(0.997\)) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(r_{xy} = r_{uv}\) (same value, awrt \(0.997\)) | B1 | Correct value stated |
| Because coding by subtracting a constant does not affect the correlation coefficient | B1 | Must give reason |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| There is a strong positive linear correlation between a rider's qualifying lap speed and their race speed | B1 | Context required |
| Riders who go faster in qualifying also tend to go faster in the race | B1 | Must reference both variables in context |
# Question 4:
## Part (a)(i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sum u = 81.58$, $\sum v = 70.11$ | B1 | May be implied |
| $\sum u^2 = 832.0708$, $\sum v^2 = 649.0983$, $\sum uv = 733.3155$ | M1 | Correct method for $S_{uu}$, $S_{vv}$, $S_{uv}$ |
| $S_{uu} = 832.0708 - \frac{81.58^2}{10} = 166.7456$ | | |
| $S_{vv} = 649.0983 - \frac{70.11^2}{10} = 157.7089$ | | |
| $S_{uv} = 733.3155 - \frac{81.58 \times 70.11}{10} = 161.7032$ | | |
| $r_{uv} = \frac{161.7032}{\sqrt{166.7456 \times 157.7089}}$ | M1 | Correct formula |
| $r_{uv} = 0.9965$ (awrt $0.997$) | A1 | |
## Part (a)(ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $r_{xy} = r_{uv}$ (same value, awrt $0.997$) | B1 | Correct value stated |
| Because coding by subtracting a constant does not affect the correlation coefficient | B1 | Must give reason |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| There is a strong positive linear correlation between a rider's qualifying lap speed and their race speed | B1 | Context required |
| Riders who go faster in qualifying also tend to go faster in the race | B1 | Must reference both variables in context |
---4 Every year, usually during early June, the Isle of Man hosts motorbike races. Each race consists of three consecutive laps of the island's course. To compete in a race, a rider must first complete at least one qualifying lap.
The data refer to the lightweight motorbike class in 2012 and show, for each of a random sample of 10 riders, values of
$$u = x - 100 \quad \text { and } \quad v = y - 100$$
where\\
$x$ denotes the average speed, in mph, for the rider's fastest qualifying lap and\\
$y$ denotes the average speed, in mph, for the rider's three laps of the race.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | c | c | c | c | c | }
\cline { 2 - 11 }
\multicolumn{1}{c|}{} & \multicolumn{11}{c|}{Rider} \\
\cline { 2 - 11 }
\multicolumn{1}{c|}{} & $\mathbf { A }$ & $\mathbf { B }$ & $\mathbf { C }$ & $\mathbf { D }$ & $\mathbf { E }$ & $\mathbf { F }$ & $\mathbf { G }$ & $\mathbf { H }$ & $\mathbf { I }$ & $\mathbf { J }$ & \\
\hline
$\boldsymbol { u }$ & 7.88 & 13.02 & 4.29 & 2.88 & 6.26 & 7.03 & 3.60 & 11.78 & 13.15 & 11.69 & \\
\hline
$\boldsymbol { v }$ & 6.63 & 10.16 & 3.63 & 0.47 & 5.70 & 8.01 & 3.30 & 7.31 & 13.08 & 11.82 & \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Calculate the value of $r _ { u v }$, the product moment correlation coefficient between $u$ and $v$.
\item Hence state the value of $r _ { x y }$, giving a reason for your answer.
\end{enumerate}\item Interpret your value of $r _ { x y }$ in the context of this question.
\end{enumerate}
\hfill \mbox{\textit{AQA S1 2014 Q4 [7]}}