State whether distance from the river bank is a continuous or discrete variable.

On graph paper, draw a scatter diagram to show Barry’s results. Use a scale of 1 cm to represent 5 m on the x-axis and 1 cm to represent 10 cm on the y-axis.

Write down

(i)     the mean distance, $\bar x$, of the trees from the river bank;

(ii)     the mean circumference, $\bar y$, of the trees.

Plot and label the point ${\text{M}}(\bar x,{\text{ }}\bar y)$ on your graph.

Write down

(i)     the Pearson’s product–moment correlation coefficient, $r$, for Barry’s results;

(ii)     the equation of the regression line $y$ on $x$, for Barry’s results.

Draw the regression line $y$ on $x$ on your graph.

Use the equation of the regression line $y$ on $x$ to estimate the circumference of a tree that is 40 m from the river bank.

continuous     (A1)

[1 mark]

     (A1)(A1)(A1)(A1)

Notes: Award (A1) for labelled axes and correct scales; if axes are reversed award (A0) and follow through for their points. Award (A1) for at least 3 correct points, (A2) for at least 6 correct points, (A3) for all 9 correct points. If scales are too small or graph paper has not been used, accuracy cannot be determined; award (A0). Do not penalize if extra points are seen.

[4 marks]

(i)     26 (m)     (A1)

(ii)     65 (cm)     (A1)

[2 marks]

point ${\text{M}}$ labelled, in correct position     (A1)(A1)(ft)

Notes: Award (A1)(ft) for point plotted in correct position, (A1) for point labelled ${\text{M}}$ or $(\bar x,{\text{ }}\bar y)$. Follow through from their answers to part (c).

[2 marks]

(i)     $-0.988\;{\text{ }}\left( {-0.988432 \ldots } \right)$     (G2)

Note: Award (G2) for $-0.99$. Award (G1) for $-0.990$.

     Award (A1)(A0) if minus sign is omitted.

(ii)     $y =  - 0.756x + 84.7$   $(y =  - 0.756281 \ldots x + 84.6633 \ldots )$     (G2)

Notes: Award (A1) for $- 0.756x$, (A1) for $84.7$. If the answer is not given as an equation, award a maximum of (A1)(A0).

[4 marks]

regression line through their ${\text{M}}$     (A1)((ft)

regression line through their $\left( {0,85} \right)$ (accept $85 \pm 1$)     (A1)(ft)

Notes: Follow through from part (d). Award a maximum of (A1)(A0) if the line is not straight. Do not penalize if either the line does not meet the y-axis or extends into quadrants other than the first.

     If ${\text{M}}$ is not plotted or labelled, then follow through from part (c).

     Follow through from their y-intercept in part (e)(ii).

[2 marks]

$- 0.756281(40) + 84.6633$     (M1)

$= 54.4{\text{ (cm) }}(54.4120 \ldots )$     (A1)(ft)(G2)

Notes: Accept $54.5$ ($54.46$) for use of 3 sf. Accept $54.3$ from use of $-0.76$ and $84.7$.

     Follow through from their equation in part (e)(ii) irrespective of working shown; the final answer seen must be consistent with that equation for the final (A1) to be awarded.

     Do not accept answers taken from the graph.

[2 marks]