QUESTION 2. (20 marks) Blood tests are required by health department legislation to be reproducible across different pathology laboratories. Concern about the reliability of data from one pathology Lab led to a study in which blood serum samples were divided and sent to two laboratories for testing. One-half of each sample was sent to the CSL Laboratory (Lab 1) and the other sent to the Astra Zeneca laboratory (Lab 2). Measurements of specific antibodies (X,) and levels of T cells (X) were obtained, for n = 11 sample splits, from the two laboratories. da = xin - x -57 27 -54 -12 -81 -30 -42 51 -66 12 -57 do = Xie - Xia -21 -6 126 -3 45 33 -12 180 30 30 36 Here d = = 1 -28.091 Sa = [1793.29 794.78] $ 1 = 0.00062 -0.000131 38.82] . 794.78 3767.561' -0.00013 0.00029] Eigenvalues and eigenvectors of 5. are: 21 = 4047.75,el = [0.33, 0.94] 12 = 1513.10,e; = [-0.94,0.33]. a) Using the information above construct and sketch a joint 95% confidence region (ellipse) for the mean difference vector &' using the data and results above. Show all your formula and working. (4 marks) b) We are interested in whether the point, 8' = [81, 82] = [0,0]. falls outside the 95% contour/ellipse. Does it? Justify your answer. (2 marks) 2 c) At what point in two-dimensional space is the ellipse centred? (1 mark) d) Determine the half length of the major axis. (2 marks) e) Determine the half length of the minor axis. (2 marks) f) In which direction does the major axis lie? (2 marks) g) In which direction does the minor axis lie? (2 marks) h) Using the information above also construct the Bonferroni 95%% simultaneous intervals for the components of the mean difference vector 8'- (4 marks) i) Out of the 11 samples tested across the 2 laboratories can you detect any outlier samples? (1 mark) Critical t values (you choose the correct one): two(D.10) = 1.372 tin(0.05] = 1.812 too[0.025) = 2.262 two(0.0125) = 2.634 two(0.01) = 2.768 two(D.005) = 3.16 Critical F values (you choose the correct one): Fa[0.10) = 3.01 Fx[0.05) = 4.26 F2,a[0.01) = 8.00