Transform PSI and recover an additional observation

A study comparing the number of hotspots detected in Indonesia each day and the average 24-hour PSI reading in Singapore was conducted. The results over 7 days were given below:

Number of hotspots ($x$)	50	97	128	150	170	189	215
Average 24-hour PSI reading ($y$)	45	98	160	205	263	341	400

(i) Draw a scatter diagram and find the value of the product moment correlation coefficient between $x$ and $y$. Hence comment on the suitability of the linear model.3 marks

(ii) Can we conclude that an increase in the number of hotspots will lead to an increase in the PSI reading? Explain your answer.1 mark

(iii) A few alternative models, in the form of $w=ax+b$, were proposed to describe the relation between $x$ and $y$. State, with a reason, which model is the most appropriate.

Model A: $w=\ln y$; Model B: $w=\sqrt y$; Model C: $w=y^2$.2 marks

It was decided that the data for an additional day should be included in the study. After inserting the data for the additional day, it was found that the equation of the regression line of $w$ on $x$ is $w=0.082742x+2.1735$ and the equation of the regression line of $x$ on $w$ is $x=12.004w-25.106$ based on the most appropriate model in part (iii).

(iv) Find the product moment correlation coefficient between $w$ and $x$.1 mark

(v) Find the average 24-hour PSI reading on this additional day, correct to the nearest integer.3 marks

In estimating the number of hotspots when the average 24-hour PSI reading was $m$, where $100<m<200$, Tarzan used the regression line of $w$ on $x$ for the estimation and Jane used the regression line of $x$ on $w$ for the estimation. It was found that both of them had approximately the same answer. Explain why this was observed.1 mark