| Exam Board | AQA |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2006 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Bivariate data |
| Type | Draw scatter diagram from data |
| Difficulty | Easy -1.2 This is a routine S1 statistics question requiring basic scatter diagram plotting and interpretation. Part (a) is pure data plotting, (b) requires simple visual interpretation, (c) identifies outliers by inspection, and only (d)(i) involves calculation (PMCC formula with 8 data points). All techniques are standard textbook exercises with no problem-solving insight required, making it easier than average A-level maths. |
| Spec | 2.02c Scatter diagrams and regression lines2.02d Informal interpretation of correlation5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation |
| Boy | A | B | C | D | E | F | G | H | I | J |
| Freestyle ( \(\boldsymbol { x }\) seconds) | 30.2 | 32.8 | 25.1 | 31.8 | 31.2 | 35.6 | 32.4 | 38.0 | 36.1 | 34.1 |
| Backstroke ( \(y\) seconds) | 33.5 | 35.4 | 37.4 | 27.2 | 34.7 | 38.2 | 37.7 | 41.4 | 42.3 | 38.4 |
| Answer | Marks | Guidance |
|---|---|---|
| 5(a) | ||
| Scatter Diagram | B2, B1, B1 | 4 labelled points plotted, 3 labelled points plotted, 4 unlabelled points plotted |
| 2 | ||
| 5(b)(i) | ||
| Positive/linear correlation/relationship except for two unusual values/results | B1, B1 | OE, OE |
| 2 | ||
| 5(b)(ii) | ||
| 0.462 | B1 | CAO; accept 3rd/final/last value |
| 1 | ||
| 5(c) | ||
| C and D: C is likely freestyle champion, D is likely backstroke champion | B1, B1 | CAO, Style identified |
| or C is likely freestyle champion, D is likely backstroke champion | B1, B1 | |
| 2 | ||
| 5(d)(i) | ||
| \(r = 0.912\) to \(0.913\) | B3 | AWFW |
| or \(r = 0.91\) to \(0.92\) or \(0.46\) to \(0.47\) | B2 | AWFW |
| or \(r = 0.9\) | B1 | AWRT |
| Attempt at \(\Sigma x\), \(\Sigma x^2\), \(\Sigma y\), \(\Sigma y^2\), \(\Sigma xy\) | M1 | 270.4, 9188.46, 301.6, 11437.84, 10246.53 |
| or Attempt at \(S_{xx}\), \(S_{yy}\), \(S_{xy}\) | M1 | 48.94, 67.52, 52.45 |
| Attempt at a correct formula for \(r\) | m1 | |
| \(r = 0.912\) to \(0.913\) | A1 | AWFW |
| 3 | ||
| 5(d)(ii) | ||
| Boys are faster/slower at both strokes or Boys are equally good at both strokes | B1 | OE; do not accept freestyle times are proportional to backstroke times |
| 1 |
| **5(a)** |
|----------|
| Scatter Diagram | B2, B1, B1 | 4 labelled points plotted, 3 labelled points plotted, 4 unlabelled points plotted |
| | | 2 |
| **5(b)(i)** |
|----------|
| Positive/linear correlation/relationship except for two unusual values/results | B1, B1 | OE, OE |
| | | 2 |
| **5(b)(ii)** |
|----------|
| 0.462 | B1 | CAO; accept 3rd/final/last value |
| | | 1 |
| **5(c)** |
|----------|
| C and D: C is likely freestyle champion, D is likely backstroke champion | B1, B1 | CAO, Style identified |
| or C is likely freestyle champion, D is likely backstroke champion | B1, B1 | |
| | | 2 |
| **5(d)(i)** |
|----------|
| $r = 0.912$ to $0.913$ | B3 | AWFW |
| or $r = 0.91$ to $0.92$ or $0.46$ to $0.47$ | B2 | AWFW |
| or $r = 0.9$ | B1 | AWRT |
| Attempt at $\Sigma x$, $\Sigma x^2$, $\Sigma y$, $\Sigma y^2$, $\Sigma xy$ | M1 | 270.4, 9188.46, 301.6, 11437.84, 10246.53 |
| or Attempt at $S_{xx}$, $S_{yy}$, $S_{xy}$ | M1 | 48.94, 67.52, 52.45 |
| Attempt at a correct formula for $r$ | m1 | |
| $r = 0.912$ to $0.913$ | A1 | AWFW |
| | | 3 |
| **5(d)(ii)** |
|----------|
| Boys are faster/slower at both strokes or Boys are equally good at both strokes | B1 | OE; do not accept freestyle times are proportional to backstroke times |
| | | 1 |
**Total for Q5: 11**
---5 [Figure 1, printed on the insert, is provided for use in this question.]\\
The table shows the times, in seconds, taken by a random sample of 10 boys from a junior swimming club to swim 50 metres freestyle and 50 metres backstroke.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline
Boy & A & B & C & D & E & F & G & H & I & J \\
\hline
Freestyle ( $\boldsymbol { x }$ seconds) & 30.2 & 32.8 & 25.1 & 31.8 & 31.2 & 35.6 & 32.4 & 38.0 & 36.1 & 34.1 \\
\hline
Backstroke ( $y$ seconds) & 33.5 & 35.4 & 37.4 & 27.2 & 34.7 & 38.2 & 37.7 & 41.4 & 42.3 & 38.4 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item On Figure 1, complete the scatter diagram for these data.
\item Hence:
\begin{enumerate}[label=(\roman*)]
\item give two distinct comments on what your scatter diagram reveals;
\item state, without calculation, which of the following 3 values is most likely to be the value of the product moment correlation coefficient for the data in your scatter diagram.
$$0.912 \quad 0.088 \quad 0.462$$
\end{enumerate}\item In the sample of 10 boys, one boy is a junior-champion freestyle swimmer and one boy is a junior-champion backstroke swimmer.
Identify the two most likely boys.
\item Removing the data for the two boys whom you identified in part (c):
\begin{enumerate}[label=(\roman*)]
\item calculate the value of the product moment correlation coefficient for the remaining 8 pairs of values of $x$ and $y$;
\item comment, in context, on the value that you obtain.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA S1 2006 Q5 [11]}}