3. Verify rigorously the equation on page 100 of textbook Faraway, that in the case of mea- surement error in simple linear regression, the expected value of the least-squares estimator is given by 2 0' + a ,5 1: ii: 03 + 0,? + 201:5 where it is assumed that the only random variables are the errors g, and that for this formula, 0:, 03, and 015 are all the sampled versions, i.e., E631) = 61 1 n _ 1 n _ 1 \" _ 2 _ A 2 2 _ 2 _ A '71. E Ell-\"Ff: 97A) , '75 E 1151: 5) a \"ma E 1031 Wilma 5) Hint: Utilize the standard formula for the least squares estimator of [31 in simple linear regression being 2\" ( _) A i=1 131' _ 3: :9"? '3': ELMT)? 0 where the 931's in the above formula are at; 's. For observational data, 0'? is (almost) the sample variance of X4 while for a con- trolled experiment we can View it as just a numerical measure of the spread of the design. A similar distinction should be made for cov(xA,5) although in many cases, it will be reasonable to assume that this is zero. Now [31 = 20:,- f)y,~/ 2(x; f)2 and after some calculation we nd that