2017 DHS P2 Q8

A retail manager of a large electrical appliances store wants to investigate the relationship between the monthly advertising expenditure, \(x\) hundred dollars, and the monthly sales of their refrigerators, \(y\) thousand dollars. The table shows the results of the investigation.

1. The manager concludes that an increase in monthly advertising expenditure will result in an increase in the monthly sales of refrigerators. State, with a reason, whether you agree with this conclusion.[1]
2. Draw a scatter diagram to illustrate the above data. Explain why a linear model is not likely to be appropriate.[2]

It is thought that the monthly sales \(y\) thousand dollars can be modelled by one of the formulae

\(y=a+b\,{{\mathrm{e}}^{\sqrt{x}}}\) or \(y=a+b\,{{x}^{2}}\)

where \(a\) and \(b\) are constants.

3. Find, correct to \(4\) decimal places, the value of the product moment correlation coefficient between
   (A) \({{\mathrm{e}}^{\sqrt{x}}}\) and \(y\),
   (B) \({{x}^{2}}\) and \(y\).
   Explain which of \(y=a+b\,{{\mathrm{e}}^{\sqrt{x}}}\) or \(y=a+b\,{{x}^{2}}\) is the better model.[2]

Assume that the better model in part (iii) holds for part (iv).

4. The manager forgot to record the monthly advertising expenditure when the monthly sales of refrigerators was \(\$11300\). Combining this with the above data set, it is found that \(a=10.876\) and \(b=0.09906\) for the model. Find the monthly advertising expenditure that the manager forgot to record, leaving your answer to the nearest hundred.[3]