N2016 P2 Q8

A website about electric motors gives information about the percentage efficiency of motors depending on their power, measured in horsepower. Xian has copied the following table for a particular type of electric motor, but he has copied one of the efficiency values wrongly.

1. Plot a scatter diagram on graph paper for these values, labelling the axes, using a scale of \(2\text{ cm}\) to represent \(10\%\) efficiency on the \(y\)-axis and an appropriate scale for the \(x\)-axis. On your diagram, circle the point that Xian has copied wrongly.[2]

For parts (ii), (iii) and (iv) of this question you should exclude the point for which Xian has copied the efficiency value wrongly.

2. Explain from your scatter diagram why the relationship between \(x\) and \(y\) should not be modelled by an equation of the form \(y=ax+b\).[1]
3. Suppose that the relationship between \(x\) and \(y\) is modelled by an equation of the form \(y=\dfrac cx+d\), where \(c\) and \(d\) are constants. State with a reason whether each of \(c\) and \(d\) is positive or negative.[2]
4. Find the product moment correlation coefficient and the constants \(c\) and \(d\) for the model in part (iii).[3]
5. Use the model \(y=\dfrac cx+d\), with the values of \(c\) and \(d\) found in part (iv), to estimate the efficiency value (\(y\)) that Xian has copied wrongly. Give two reasons why you would expect this estimate to be reliable.[3]