Answer the questions in the attachments below.

A university wishes to analyse the performance of its students on a particular degree course. It records the scores obtained by a sample of 12 students at entry to the course, and the scores obtained in their final examinations by the same students. The results are as follows: Student A B C D E F G H K L Entrance exam score x (%) 86 53 71 62 79 24 90 55 58 72 Finals paper score y (%) 75 74 68 70 75 78 90 85 62 70 [x=836 _ y =867 _x2 = 60,016 _ y? = 63,603 )(x-X)(y- y) =1,122 (i) Calculate the fitted linear regression equation of y on x. (ii) Assuming the full normal model, calculate an estimate of the error variance o and obtain a 90% confidence interval for oz. (iii) By considering the slope parameter, formally test whether the data is positively correlated. (iv) Calculate a 95% confidence interval for the mean finals paper score corresponding to an individual entrance score of 53. (v) Test whether this data could come from a population with correlation coefficient equal to 0.75. (vi) Calculate the proportion of variation explained by the model. Hence, comment on the fit of the model