1. Fifty people participated in a medical study. Each participant contributed two eye measurements (labeled y ). Thus the study had a total of 100 measurements on y from the 50 participants. It is suspected that the two eye measurements from the same person are correlated. Witho'ut sufficient knowledge of the study background, a data analyst performed a multiple linear regression analysis using a model with three regressors, x4, X3, x3, and assuming that all 100 measurements are statistically independent (i.e., no correlation between participants and within each participant). The model can be expressed as E(y |x;,X2,x3) = o + f1x; + f2x7 + [3x3 . Thus the data analyst performed OLS (i.e., ordinary least-squares) estimation for all regression coefficients. a) The data analyst assumed that all 100 measurements are statistically independent and used the OLS estimator to generate test for testing significance of x; at the target significance level . Is the type | error probability of this t test no larger than the targeted level? Provide mathematical proof for your answer. [10 points] b) Later the data analyst was told that the statistical independence assumption did not hold. The data analyst was given the variance-covariance matrix V = 57 1 025 , N (0.25 1 )for each pair of eye measurements of each participant, where a2 is unknown and needs to be estimated. (b1) Should the analyst adjust estimation and statistical testing for the regression coefficient vector B = (B, f1, B2, B3)' ? Provide mathematical proof for your answer. [10 points] (b2) Construct an unbiased estimator for o and show unbiasedness. [10 points] (b3) Construct the best linear unbiased estimator for 5, and its 95% confidence interval with mathematical proof. [10 points] (b4) Construct a statistically valid test for testing H,: 3; = 0 and provide mathematical proof for its validity. [10 points]