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.
| \(x\) | \(40\) | \(45\) | \(50\) | \(55\) | \(60\) |
| \(y\) | \(22\) | \(20\) | \(18\) | \(17\) | \(16\) |
Abi decides to model the data using the line \(y=35-\frac13x\).
On the grid opposite
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.[1]
Calculate the sum of the squares of the residuals for Abi’s line.[1]
Explain why, in general, the sum of the squares of the residuals rather than the sum of the residuals is used.[1]
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 \(1\).
State, with a reason, which of the two models, Abi’s or Bhani’s, gives a better fit.[1]
State the coordinates of the point that the least squares regression line must pass through.[1]
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.[3]
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.[2]
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?[1]