2. A small company has four workshops, each producing hand-crafted furniture. The amount of items produced in each of two months for the workshops is given below. Workshop 1 1948 Workshop 2 45 23 Workshop 3 70 Workshop 4 93 127 62 a) Consider a linear model for the data where the number of items produced by each workshop has different means u + Qi, i = 1, 2, 3, 4, but the same variance, and the observations can be assumed to be uncorrelated. Using B = (, Q1, Q2, Q3, 04)" and an appropriate design matrix, write down the model in matrix notation, and state the normal equations. b) Write down a solution to the normal equations for this dataset. You may use the following generalized inverse of X+X: (x+x) 0 0 0 0.5 0 0 0 0 0 0 0 0 0 0.5 0 0 0.5 0 0 0 0 0 0 0.5 c) Are the following quantities linearly estimable, and give your reasons? i) 201 - 02) ii) u + Q1 - 02 iii) x + (a1 +22)/2 Note: One convenient way of verifying linear estimability is to obtain a basis vector for the null space of the design matrix, and then this is orthogonal to any linear estimable function. d) For the quantities in (c) that are linearly estimable use the Gauss-Markov theorem to obtain the Best Linear Unbiased Estimates. e) Staff in workshops 3 and 4 have had advanced training whereas those in Workshops 1 and 2 have not. The management claims that production will increase by 100% if advanced training is given for the workforce. Suggest a suitable linear model with full rank design matrix for this situation, with parameter B denoting the mean number of items produced by Workshop 1. Hence, obtain the least squares estimate of the parameter B under the model. 2. A small company has four workshops, each producing hand-crafted furniture. The amount of items produced in each of two months for the workshops is given below. Workshop 1 1948 Workshop 2 45 23 Workshop 3 70 Workshop 4 93 127 62 a) Consider a linear model for the data where the number of items produced by each workshop has different means u + Qi, i = 1, 2, 3, 4, but the same variance, and the observations can be assumed to be uncorrelated. Using B = (, Q1, Q2, Q3, 04)" and an appropriate design matrix, write down the model in matrix notation, and state the normal equations. b) Write down a solution to the normal equations for this dataset. You may use the following generalized inverse of X+X: (x+x) 0 0 0 0.5 0 0 0 0 0 0 0 0 0 0.5 0 0 0.5 0 0 0 0 0 0 0.5 c) Are the following quantities linearly estimable, and give your reasons? i) 201 - 02) ii) u + Q1 - 02 iii) x + (a1 +22)/2 Note: One convenient way of verifying linear estimability is to obtain a basis vector for the null space of the design matrix, and then this is orthogonal to any linear estimable function. d) For the quantities in (c) that are linearly estimable use the Gauss-Markov theorem to obtain the Best Linear Unbiased Estimates. e) Staff in workshops 3 and 4 have had advanced training whereas those in Workshops 1 and 2 have not. The management claims that production will increase by 100% if advanced training is given for the workforce. Suggest a suitable linear model with full rank design matrix for this situation, with parameter B denoting the mean number of items produced by Workshop 1. Hence, obtain the least squares estimate of the parameter B under the model