9758/2019/P2/Q10

Abi and Bhani find the fuel consumption for a car driven at different constant speeds. The table shows the fuel consumption, $y$ kilometres per litre, for different constant speeds, $x$ kilometres per hour.

(i) Abi decides to model the data using the line $y=35-\frac{1}{3} x$.

(a) On the grid opposite

• draw a scatter diagram of the data,
• draw the line $y=35-\frac{1}{3} x$.



(b) For a line of best fit $y=\mathrm{f}(x)$, the residual for a point $(a, b)$ plotted on the scatter diagram is the vertical distance between $(a, \mathrm{f}(a))$ and $(a, b)$. Mark the residual for each point on your diagram.



(c) Calculate the sum of the squares of the residuals for Abi’s line.



(d) Explain why, in general, the sum of the squares of the residuals rather than the sum of the residuals is used.



Bhani models the same data using a straight line passing through the points $(40,22)$ and $(55,17)$. The sum of the squares of the residuals for Bhani’s line is 1 .

(ii) State, with a reason, which of the two models, Abi’s or Bhani’s, gives a better fit.



(iii) State the coordinates of the point that the least squares regression line must pass through.



(iv) Use your calculator to find the equation of the least squares regression line of $y$ on $x$. State the value of the product moment correlation coefficient.



(v) Use the equation of the regression line to estimate the fuel consumption when the speed is 30 kilometres per hour. Explain whether you would expect this value to be reliable.



(vi) Cerie performs a similar experiment on a different car. She finds that the sum of the squares of the residuals for her line is 0 . What can you deduce about the data points in Cerie’s experiment?