6 (8pts)2x4 The following represents observations (m1 , yl), (932, 3:3), . . . , (mm y\"), n = 8, of two variables X, Y: We've obtained the regression t via the nonlinear model Y} = :12? + a + 5,, where a is unknown, and the random noise 5,; has distribution N(U,02). We'll view the above values of 22,; as xed, i.e., they are not random, but Yg's are random and independent. For the above t on the right, we derived a least-squares estimator a for the unknown a, and aiready computed E: = 22.95 on the above data (so you would not have to). (a) Derive the least-squares estimator a of a, i.e., the a minimizing the squared-error on the data (21,1;1), . .. (mmyn). No need to argue second order conditions (i.e., the squared error is convex in terms of a). (b) Derive the distribution of ('1, and be explicit about its mean and variance. (c) As stated earlier, the right least-squares formula yields 6:. = 22.95 for the above data. Using this value of ti, derive a 98% condence interval for 0., assuming a2 is known. No need to compute the exact interval, but express it in terms of critical values of a known distribution (and other necessary quantities). (cl) Suppose we don't know :52. Which of the following two quantities (i) and (ii) is a more reasonable estimate for org? Give a brief reason why. (i) $20:- Y)\". (ii) $201- -3; a)? i=1