Draw a scatter diagram to show the above data. Use a scale of \(2{\text{ cm}}\) to represent \(10\% \) on the x-axis and a scale of \(2{\text{ cm}}\) to represent \(10\) thermal units on the y-axis.

Write down
(i)     the mean percentage moisture content, \(\bar x\) ;
(ii)    the mean heat output, \(\bar y\) .

Plot the point \((\bar x{\text{, }}\bar y)\) on your scatter diagram and label this point M .

Write down the product-moment correlation coefficient, \(r\) .

The equation of the regression line \(y\) on \(x\) is \(y = - 0.470x + 83.7\) . Draw the regression line \(y\) on \(x\) on your scatter diagram.

The equation of the regression line \(y\) on \(x\) is \(y = - 0.470x + 83.7\) . Estimate the heat output in thermal units of a \(1{\text{ kg}}\) block of wood that has \(25\% \) moisture content.

The equation of the regression line \(y\) on \(x\) is \(y = - 0.470x + 83.7\) . State, with a reason, whether it is appropriate to use the regression line \(y\) on \(x\) to estimate the heat output in part (f).

     (A1) for correct scales and labels
     (A3) for all ten points plotted correctly
     (A2) for eight or nine points plotted correctly
     (A1) for six or seven points plotted correctly     (A4)

Note: Award at most (A0)(A3) if axes reversed.

[4 marks]

(i)     \(\bar x = 42\)     (A1)

(ii)    \(\bar y = 64\)     (A1)

[2 marks]

\((\bar x{\text{, }}\bar y)\) plotted on graph and labelled, M     (A1)(ft)(A1)

Note: Award (A1)(ft) for position, (A1) for label.

[2 marks]

\( - 0.998\)    (G2)

Note: Award (G1) for correct sign, (G1) for correct absolute value.

[1 mark]

line on graph (A1)(ft)(A1)

Notes: Award (A1)(ft) for line through their M, (A1) for approximately correct intercept (allow between \(83\) and \(85\)). It is not necessary that the line is seen to intersect the \(y\)-axis. The line must be straight for any mark to be awarded.

[2 marks]

\(y =  - 0.470(25) + 83.7\)     (M1)

Note: Award (M1) for substitution into formula or some indication of method on their graph. \(y =  - 0.470(0.25) + 83.7\) is incorrect.

\( = 72.0\) (accept \(71.95\) and \(72\))     (A1)(ft)(G2)

Note: Follow through from graph only if they show working on their graph. Accept \(72 \pm 0.5\) .

[2 marks]

Yes since \(25\% \) lies within the data set and \(r\) is close to \( - 1\)     (R1)(A1)

Note: Accept Yes, since \(r\) is close to \( - 1\)

Note: Do not award (R0)(A1).

[2 marks]

The great majority of candidates found this question to be a good start to the paper. The common errors were (1) incorrect scales being used; SI units are standard in this course and candidates are expected to know the difference between centimetres and millimetres (2) the lack of \(r\) on the GDC (3) not knowing that the regression line \(y\) on \(x\) passes through the mean point and (4) not realising that the value of \(r\) determines the validity of using the regression line \(y\) on \(x\) .

The great majority of candidates found this question to be a good start to the paper. The common errors were (1) incorrect scales being used; SI units are standard in this course and candidates are expected to know the difference between centimetres and millimetres (2) the lack of \(r\) on the GDC (3) not knowing that the regression line \(y\) on \(x\) passes through the mean point and (4) not realising that the value of \(r\) determines the validity of using the regression line \(y\) on \(x\) .

The great majority of candidates found this question to be a good start to the paper. The common errors were (1) incorrect scales being used; SI units are standard in this course and candidates are expected to know the difference between centimetres and millimetres (2) the lack of \(r\) on the GDC (3) not knowing that the regression line \(y\) on \(x\) passes through the mean point and (4) not realising that the value of \(r\) determines the validity of using the regression line \(y\) on \(x\) .

The great majority of candidates found this question to be a good start to the paper. The common errors were (1) incorrect scales being used; SI units are standard in this course and candidates are expected to know the difference between centimetres and millimetres (2) the lack of \(r\) on the GDC (3) not knowing that the regression line \(y\) on \(x\) passes through the mean point and (4) not realising that the value of \(r\) determines the validity of using the regression line \(y\) on \(x\) .

The great majority of candidates found this question to be a good start to the paper. The common errors were (1) incorrect scales being used; SI units are standard in this course and candidates are expected to know the difference between centimetres and millimetres (2) the lack of \(r\) on the GDC (3) not knowing that the regression line \(y\) on \(x\) passes through the mean point and (4) not realising that the value of \(r\) determines the validity of using the regression line \(y\) on \(x\) .

The great majority of candidates found this question to be a good start to the paper. The common errors were (1) incorrect scales being used; SI units are standard in this course and candidates are expected to know the difference between centimetres and millimetres (2) the lack of \(r\) on the GDC (3) not knowing that the regression line \(y\) on \(x\) passes through the mean point and (4) not realising that the value of \(r\) determines the validity of using the regression line \(y\) on \(x\) .

The great majority of candidates found this question to be a good start to the paper. The common errors were (1) incorrect scales being used; SI units are standard in this course and candidates are expected to know the difference between centimetres and millimetres (2) the lack of \(r\) on the GDC (3) not knowing that the regression line \(y\) on \(x\) passes through the mean point and (4) not realising that the value of \(r\) determines the validity of using the regression line \(y\) on \(x\) .