N2013 P2 Q10

1. Sketch a scatter diagram that might be expected when \(x\) and \(y\) are related approximately as given in each of the cases (A), (B) and (C) below. In each case your diagram should include \(6\) points, approximately equally spaced with respect to \(x\), and with all \(x\)- and \(y\)-values positive. The letters \(a\), \(b\), \(c\), \(d\), \(e\) and \(f\) represent constants.[3]

(A) \(y=a+bx^2\), where \(a\) is positive and \(b\) is negative.

(B) \(y=c+d\ln x\), where \(c\) is positive and \(d\) is negative.

(C) \(y=e+\frac f x\), where \(e\) is positive and \(f\) is negative.

A motoring website gives the following information about the distance travelled, \(y\text{ km}\), by a certain type of car at different speeds, \(x\text{ km h}^{-1}\), on a fixed amount of fuel.

2. Draw the scatter diagram for these values, labelling the axes.[1]
3. Explain which of the three cases in part (i) is the most appropriate for modelling these values, and calculate the product moment correlation coefficient for this case.[2]
4. It is required to estimate the distance travelled at a speed of \(110\text{ km h}^{-1}\). Use the case that you identified in part (iii) to find the equation of a suitable regression line, and use your equation to find the required estimate.[3]