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 60 62 79 66 84 90 55 58 72 Finals paper score y (%) 75 60 74 68 70 75 78 90 85 60 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.Five students have compared their scores in some practice papers that they sat before their exam. Their marks were as follows: Student 1 72, 75, 62, 71, 60, 59 Student 2 78, 82, 64, 72 Student 3 90, 78, 67, 71, 83 Student 4 80, 77, 76, 81, 64 Student 5 95, 88, 62 Consider the model: i = 1,2,3,4,5 j = 1,2, ..., n; eij ~ N(0,62) where y;; is the score of the ith student on the j th practice paper and n, is the number of papers taken by student i. The e; are independent and identically distributed and Ent; =0. (i) Calculate the least squares estimates of / and t;, i =1,2,3,4,5. (ii) Perform an analysis of variance on these results stating clearly the null hypothesis and your conclusion. (iii) Calculate a 95% confidence interval for the underlying common standard deviation for all students assuming that the null hypothesis holds true.(a) In a perfectly competitive market, a firm's average revenue and cost functions are given as follows: AR = * Q - B where a, B are constants and Q is the output AC = 0 - B AR is the average revenue and AC is the average cost. Q on the basis of the functions given above, determine: Total revenue function. (2 marks) 300 Total cost function. (2 marks) Total break-even output level. (2 marks)