QUESTION 3 [33 Marks] Deoxyribonucleic acid (DNA) is the hereditary material found in most organisms. A genome is an organism's complete set of DNA, including all of its genes. Each genome contains all the information needed to build and maintain that organism. Consider the following data on genome size measured in pictograms (trillionths of a gram) of DNA per haploid cell in several large groups of crustaceans. Type of Crustacean Amphipods Barnacles Branchiopods 27.00 0.67 0.19 50.91 0.90 64.42 Total 142.53 1.57 0.19 Let vij denote the DNA content of the th crustacean of the ith type, th taking values 1, 2, and 3 for amphipods, barnacles, and branchiopods, respectively, and j = 1, 2, ..., "; , where n, is the number of observations of the th type. Assume the following model, Vij = u + a; + eij. 3.1 Write down the six observations in terms of the equation of the model. (6) 3.2 Rewrite these equations in matrix form, y = XB + c. (6) 3.3 Consider the following matrices 13 27 G3 = 36 -21 2 0 35 and G4 = 0 -25 0 3 -25 59 to be the generalised inverses of X'X. 3.3.1 Find the solutions to the normal equations using each of the generalised inverses. (6) 3.3.2 Find two linear combinations of these solutions that are the same and two that are dif- ferent. (4) 3.3.3 Show that y and the sum of squares regression (SSR) are the same for the two gener- alised inverses. (6) 3.3.4 Show that a2 - as has the same variance when derived from G3 and G4