| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2005 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Bivariate data |
| Type | Calculate r from raw bivariate data |
| Difficulty | Moderate -0.3 This is a standard S1 correlation calculation requiring systematic computation of Σx, Σy, Σx², Σy², Σxy and application of the PMCC formula. While tedious with 9 data pairs, it follows a routine algorithmic procedure with no conceptual challenges beyond formula recall and careful arithmetic—slightly easier than average due to its purely mechanical nature. |
| Spec | 2.02c Scatter diagrams and regression lines2.02d Informal interpretation of correlation5.08a Pearson correlation: calculate pmcc |
| Monday | \(\mathbf { 1 }\) | \(\mathbf { 2 }\) | \(\mathbf { 3 }\) | \(\mathbf { 4 }\) | \(\mathbf { 5 }\) | \(\mathbf { 6 }\) | \(\mathbf { 7 }\) | \(\mathbf { 8 }\) | \(\mathbf { 9 }\) | \(\mathbf { 1 0 }\) |
| \(\boldsymbol { x }\) | 2 | 6 | 9 | 18 | 1 | 3 | 7 | 12 | 13 | 4 |
| \(\boldsymbol { y }\) | 97 | 103 | 136 | 245 | 121 | 78 | 145 | 128 | 141 | 312 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Takings appear to increase slightly as air temperature increases; weak positive (linear) correlation between air temperature and takings | B1 | OE; comments on ranges of values of \(x\) and \(y \Rightarrow\) B0 |
| One (or two) unusual results | B1 | OE |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Monday 10 | B1 | CAO; accept point \((4, 312)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(r = 0.817\) to \(0.818\) | B3 | AWFW; for attempts at \(\Sigma x\), \(\Sigma x^2 \times 5\) or \(S_{xx} \times 3\) — M1; for attempted use of correct formula for \(r\) — M1; for answer — A1 |
| If Monday 4 identified in (b): \(r = 0.0156\) to \(0.0157\) | — | scores M2 |
| If no Monday removed: \(r = 0.318\) to \(0.319\) | — | scores M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Temperature at another time; number of other/competing stalls; month/time of year; rainfall/snow; publicity | E1 | Or a sensible alternative; number of customers \(\Rightarrow\) E0; weather \(\Rightarrow\) E0; population of town \(\Rightarrow\) E0 |
## Question 1:
### Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| Takings appear to increase slightly as air temperature increases; weak positive (linear) correlation between air temperature and takings | B1 | OE; comments on ranges of values of $x$ and $y \Rightarrow$ B0 |
| One (or two) unusual results | B1 | OE |
### Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| Monday 10 | B1 | CAO; accept point $(4, 312)$ |
### Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| $r = 0.817$ to $0.818$ | B3 | AWFW; for attempts at $\Sigma x$, $\Sigma x^2 \times 5$ or $S_{xx} \times 3$ — M1; for attempted use of correct formula for $r$ — M1; for answer — A1 |
| If Monday 4 identified in (b): $r = 0.0156$ to $0.0157$ | — | scores M2 |
| If no Monday removed: $r = 0.318$ to $0.319$ | — | scores M1 |
### Part (d)
| Answer | Mark | Guidance |
|--------|------|----------|
| Temperature at another time; number of other/competing stalls; month/time of year; rainfall/snow; publicity | E1 | Or a sensible alternative; number of customers $\Rightarrow$ E0; weather $\Rightarrow$ E0; population of town $\Rightarrow$ E0 |
---1 Each Monday, Azher has a stall at a town's outdoor market. The table below shows, for each of a random sample of 10 Mondays during 2003, the air temperature, $x ^ { \circ } \mathrm { C }$, at 9 am and Azher's takings, £y.
\begin{center}
\begin{tabular}{ | c | r | r | r | r | r | r | r | r | r | r | }
\hline
Monday & \multicolumn{1}{|c|}{$\mathbf { 1 }$} & \multicolumn{1}{c|}{$\mathbf { 2 }$} & \multicolumn{1}{c|}{$\mathbf { 3 }$} & \multicolumn{1}{c|}{$\mathbf { 4 }$} & \multicolumn{1}{c|}{$\mathbf { 5 }$} & $\mathbf { 6 }$ & \multicolumn{1}{c|}{$\mathbf { 7 }$} & \multicolumn{1}{c|}{$\mathbf { 8 }$} & \multicolumn{1}{c|}{$\mathbf { 9 }$} & \multicolumn{1}{c|}{$\mathbf { 1 0 }$} \\
\hline
$\boldsymbol { x }$ & 2 & 6 & 9 & 18 & 1 & 3 & 7 & 12 & 13 & 4 \\
\hline
$\boldsymbol { y }$ & 97 & 103 & 136 & 245 & 121 & 78 & 145 & 128 & 141 & 312 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item A scatter diagram of these data is shown below.\\
\includegraphics[max width=\textwidth, alt={}, center]{7faa4a2d-f5cc-4cc3-a3a9-5d8290ceabdc-2_901_1068_1078_447}
Give two distinct comments, in context, on what this diagram reveals.
\item One of the Mondays is found to be Easter Monday, the busiest Monday market of the year. Identify which Monday this is most likely to be.
\item Removing the data for the Monday you identified in part (b), calculate the value of the product moment correlation coefficient for the remaining 9 pairs of values of $x$ and $y$.
\item Name one other variable that would have been likely to affect Azher's takings at this town's outdoor market.\\
(l mark)
\end{enumerate}
\hfill \mbox{\textit{AQA S1 2005 Q1 [7]}}