Question

4. (25 points) A clothing manufacturer has three types of machines that create rolls of fabric from yarn. Further, there are three operators for each machine. For instance, for machine A, Adam, Anna, and Arnold are the operators. For machine B, Bill Brenda, and Barbara are the operators. For machine C, Caren, Charlie, and Chris are the operators. The plamt engineer is interested in studying the variation in the strength of the fabric produced 4 pieces of fabric from each operator and measures the breaking strength of each sampled fabric piece. across machines and operators. She samples п %3 (a) Write out the statistical model, clearly stating all parameters, and ranges of subscripts (and standard deviations) (b) The means machine/operator. Give the parameter estimates for all model parameters. given in the following table for each are Operator Machine A 25.0 (2.0) 28.0 (3.0) 22.0 (1.0 Machine B Machine C 18.0 (1.0 25.0 (3.0) 17.0 (2.0 31.0 (3.0 30.0 (2.0) 29.0 (2.0 2 3

Answer 1

This is two way ANQVA model as it controls variation from two sides namely from operators and machines. The model of two way ANQVA is given below, y_ij = mu + alpha_i + beta_j + epsilon_ijk Where, y_ij = is the observation corresponding to i^th level of factor A and j^th level of factor B. mu = Common mean effect. alpha_i = effect due to i^th level of factor A. Where i = 1, 2 ellipsis P beta_j = effect due to j^th level of factor B Where j = 1, 2 ellipsis q epsilon_ij = random error \r\n Let us assume treatments as factor A and operators as factor B. Consider the following table which includes the mean of each machine and each operator Estimate of mu = y^bar = Common mean = 225/9 = 25 Estimate of alpha_i = 74 + 83 + 68/3 -25 = 75 - 25 alpha_i = 50 Estimate of beta_j = (75 + 90