Page 1 625.661 Statistical Models and Regression Test 1 for Modules 1, 2, 3, 4 Do all the problems (2 pages) below. Provide intermediate steps and state the assumptionls) for each intermediate step of your work. 1. In a multiple linear regression analysis, (31,-, x1i,x2i),i = 1, ...n , are statistically independent and satisfy the model (M1) given by: y=50+51x1+l82x2+5. where the response variable y is continuous, the regressor vector X = (x1,x2)' has mean vector (juxl, llxz)' and positive definite variance-covariance matrix 2x , the random errors 8,- conditional on X,- are statistically independent and normally distributed with mean zero and variance 0'2 which does not depend on X. a) b) Consider the case that the value of 02 is known; that is, it is given. 1) Construct a statistical test for testing H0: 31 = z = 0, using the known 02, and the a-level rejection region of the test. [5 points] 2) Construct a statistical test for testing H0: 31 = [32, using the known 02, and the cr- level rejection region of the test. [5 points] 3) Are there any differences in testing between al) and 32) above? why or why not? [5 points] Consider the case that the value of 02 is unknown; that is, it needs to be estimated. 1) Construct a statistical test for testing H0:B1 = z = 0 and the at-level rejection region of the test. [5 points] 2) Construct a statistical test for testing each individual regressor; that is, for the ith regressor (i = 1, 2), Hm: Bi = 0 and the a-Ievel rejection region of the test. [5 points] 3) In b2) above, derive the probability of falsely rejecting at least one H0,- , i = 1, 2. In b1) above, derive the probability of falsely rejecting at least one H0,- , i = 1,2. [7 points] If ,80 may or may not be zero in the model M1, construct analysis-of-variance (or sum-of- squares) table for this multiple linear regression analysis. [3 points]