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 specic antibodies (XI) and levels of T cells (X2) were obtained, for n = 11 sample splits, from the two laboratories. d1 =x1-1x2-1 \"WE Emu-1mm (i: 31 [_28.09 s = 1793.29 794.78 s1=[ 0.00062 0.00013 38.82 d 794.78 3767.56' '1 0.00013 0.00029" Eigenvalues and eigenvectors of Sd are: .11 = 4047.75,e; = [0.33.094] 712 = 1513.10, 85 = [0.94, 0.33]. a) Using the information above construct and sketch a joint 90% confidence region (ellipse) for the mean difference vector 5' using the data and results above. Show all your formula and working. (4 marks) b) We are interested in whether the point, = [51: 52] = [0: 0] , falls outside the 90% contour/ ellipse. Does it? Justify your answer. (2 marks) 0) At what point in two-dimensional space is the ellipse centred? (1 mark) (1) 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 90% simultaneous intervals for the components of the mean difference vector 8'. (4 marks) 1) 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): t10(0.10) = 1.372 t10(0.05) = 1.812 t10(0.025) = 2.262 t10(0.0125) = 2.634 t10(0.01) = 2.768 t10(0.005) = 3.16 Critical F values (you choose the correct one): F2,9(0.10) = 3.01 F2,9(0.05) = 4.26 F2,9(0.01) = 8.00