4 (6 pts) 1, 1, 1, 1, 2 Let X and Y denote the monthly prots (in units of $1M) of two edgling oil companies, Drill, and DrillMore. There is reason to believe that prots are linearly related as Y = aX + b + E, for a noise term 6 ~ N(O, 02), 6 independent of X, where a and b are both unknown. Suppose we know the prots x1, x2, . . . ,xn of Drill for the last n months, i.e., xi's are constants, and we are to further collect prot data Y1, Y2, . . . , Yn for DrillMore over the same months. From these we would estimate a and b as d = Czy/Cm, and b=Y&-a'c. a Show that I; = l '3': cin: for some constants q's depending just on the $33. '1. 1 11. Remark: the nest questions don't require answering (a). (b) Explain briey why the fact that 13 = 1 2:1 ciY; imply that 5 must be Gaussian. n. (c) What is the expectation of Y when Drill's prot X = 0? Now, we know i) is unbiased for b and has variance %(1 + 01:21)\"? Suppose an investor is wondering whether, when Drill makes no prot (i.e. when X = 0), one could still expect positive prots out of DrillMore. This can be set up as hypotheses H0 : b = b0 vs H1 : b > b0. ((1) What should be above be? (e) Assume a2 is known. Derive an a level test (in terms of 'z' critical values) which uses a test statistic involving b