Choose the regression direction for an inverse prediction

A new drug developed to reduce a smoker’s reliance on nicotine may reduce one’s pulse rate to dangerously low levels. To investigate the drug’s effect on pulse rate, different dosages of the drug were administered to six smokers, and the decrease in each smoker’s pulse rate was recorded 30 minutes later.

Smoker	1	2	3	4	5	6
Drug dosage ($x\ \mathrm{cm}^3$)	2.0	1.5	3.0	2.5	4.0	3.5
Decrease in pulse rate ($y$ beats per minute)	14	9	18	16	21	20

(i) Draw a scatter diagram to illustrate the data.1 mark

(ii) Calculate the product moment correlation coefficient between $x$ and $y$, and explain why its value does not necessarily mean that the best model for the relationship between $x$ and $y$ is $y=a+bx$.2 marks

(iii) It is proposed that the decrease in pulse rate can be modelled by the formula $y=\ln(cx^d)$, where $c$ and $d$ are constants. Explain why this is a better model.1 mark

(iv) Calculate the least-squares estimates of $c$ and $d$ for the model in (iii).2 marks

(v) Use a regression line to give the best estimate that you can of the dosage of drug administered when the decrease in pulse rate is 25 beats per minute. Comment on the reliability of your answer.3 marks