Recover a missing score and compare model shapes

Eight randomly selected students were asked to indicate the number of hours they spent studying Mathematics per week during the June holiday. The corresponding Mathematics Mid-year examination scores marked out of 100 marks of the eight students were recorded. The eight pairs of data are given in the table below.

Time spent ($x$ hours)	2.0	2.4	3.0	3.8	4.8	7.2	8.2	9.4
Mathematics score ($y$)	10	19	$p$	47	64	78	76	84

(i) Given that the regression line of $y$ on $x$ is $y=1.9263+9.6223x$, find the value of $p$.3 marks

(ii) Draw a scatter diagram to illustrate the data. Calculate the product moment correlation coefficient between $x$ and $y$.2 marks

(iii) Calculate the least squares estimates of $a$ and $b$ for the model $y=a+b\ln x$. Find the product moment correlation coefficient between $\ln x$ and $y$.2 marks

(iv) Using your answers to parts (i) and (ii), explain which of $y=1.9263+9.6223x$ or $y=a+b\ln x$ is the better model.2 marks

(v) Using the appropriate regression line, estimate the Mathematics Mid-year examination score of a student who spent 5.0 hours studying Mathematics during the June holiday. Comment on the reliability of your estimate.2 marks