Help in solving the following questions.

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; e; - N(0.02) where yo is the score of the ith student on the / th practice paper and n, is the number of papers taken by student i. The ey are independent and identically distributed and Ent; =0. (1) Calculate the least squares estimates of / and r;. 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.3. [15 points] Consider the following market equilibrium model. The demand and supply functions are 1 = a- bP - exp (cP) + dIn (Yo) , Q. = e+fP+gpo, respectively. P is the market price, Yo is income, and In (.) is the logarithmic function with the natural base. Assume that a, b, c, d, e, f, g, and o are positive constants. (a) [5 points] In the equilibrium in which Q = Qa = Qs, we may set up a two-equation system: F' (P.Q;Yo) = a-bP - exp (cP) + din (Yo) - Q=0 (20) F2 (P.Q; Yo) = e+fP+ gpo - Q=0, (21) 3 Show that in this two-equation system the conditions of the implicit function theorem hold