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4. Consider a design with factors A and B, under which there are a levels for...
Consider a factorial design model with 2 levels of Factor A, 3 levels of Factor B,...
Consider a factorial design model with 2 levels of Factor A, 3 levels of Factor B, and 2 observations at each combination of factor levels. Write this model in matrix notation as Y = X8+ €. Compute the matrices X'X and X'Y and use them to derive the least-squares estimators of all appropriate parameters from the normal equations XXB = X'Y Note: Do not write the matrices for a general factorial design model. Consider the particular factorial design described here.
Consider a four-factor factorial experiment where factor A is at a levels, factor B is at...
Consider a four-factor factorial experiment where factor A is at a levels, factor B is at b levels, factor C is at c levels, factor D is at d levels, and there are n replicates. Write down the sums of squares, the degrees of freedom, and the expected mean squares for the following cases. Assume the restricted model for all mixed models. You may use a computer package such as Minitab. (d) A and B are fixed and C and...
3. Consider the two-factor model with interaction Suppose that there are a and b levels of...
3. Consider the two-factor model with interaction Suppose that there are a and b levels of the factors respectively. Now consider the set of equations (a) Show that the equations are not redundant. (b) Show that these equations are equivalent to the hypothesis of no interaction. (c) Thereby calculate the rank of the hypothesis of no interaction. (d) Show that the hypothesis is testable, provided there exists at least one sample from each combination of factor levels.
4. In a process development study on yield, four factors were studied, each at two levels: time (A), concentration (B),...
4. In a process development study on yield, four factors were studied, each at two levels: time (A), concentration (B), pressure (C) and temperature (D). A single replicate of a 24 design was run, and the resulting are shown in the Table below a. Construct a normal probability plot of the effect estimates. Which factors appear to have large effects? b. Conduct an analysis of variance using the normal probability plot in part (a) for guidance in forming an error...
6.5 An engineer is interested in the effects of cutting speed (A), tool geometry (B), and cutting...
Solve parts b and d by using SPSS
6.5 An engineer is interested in the effects of cutting speed (A), tool geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three replicates of a 2 factorial design are run. The results are as follows: Replicate Treatment B C Combination 22 31 32 43 35 34 50 55 47 46 44 40 37 36 60 50 54...
Problem #4 (25). In class. An engineer is interested in the effects of cutting speed (A), too! geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each fac...
Problem #4 (25). In class. An engineer is interested in the effects of cutting speed (A), too! geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three replicates of a 23 factorial design are run. The results are as follows AİB Treatment combination Replicate 31 43 34 47 45 37 25 29 32 35 46 38 36 54 47 ас 41 abc B. Suppose that these...
Two levels of each factor are chosen, and three replicates of a 23 factorial design are run. The ...
Two levels of each factor are chosen, and three replicates of a 23 factorial design are run. The results are as follows: Treatment Replicate ABCCombination I II III (1) 22 31 25 a 32 43 29 b 35 34 50 ab 55 47 46 c 44 45 38 ac 40 37 36 bc 60 50 54 abc 39 41 47. Several times we have used the hierarchy principle in selecting a model; that is, we have included nonsignificant lower order...
1. An engineer is interested in the effects of cutting speed (A), tool geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three re...
1. An engineer is interested in the effects of cutting speed (A), tool geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three replicates of a 23 factorial design are run. The results are as follows: Treatment Replicate A B CCombinationI 22 31 32 43 35 34 55 47 44 45 40 37 60 50 54 39 41 25 29 50 46 38 36 ac bc...
Solve parts b and d by using SPSS 6.5 An engineer is interested in the effects...
Solve parts b and d by using SPSS
6.5 An engineer is interested in the effects of cutting speed (A), tool geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three replicates of a 2 factorial design are run. The results are as follows: Replicate Treatment B C Combination 22 31 32 43 35 34 50 55 47 46 44 40 37 36 60 50 54...
Problem #4 (25). In class. An engineer is interested in the effects of cutting speed (A), too! geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each fac...
Problem #4 (25). In class. An engineer is interested in the effects of cutting speed (A), too! geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three replicates of a 23 factorial design are run. The results are as follows: Treatment combination Replicate 31 43 34 47 45 37 50 41 25 29 50 32 35 ab 40 ас bc abc 54 47 A. What is...
Consider a factorial design model with 2 levels of Factor A, 3 levels of Factor B, and 2 observations at each combination of factor levels. Write this model in matrix notation as Y = X8+ €. Compute the matrices X'X and X'Y and use them to derive the least-squares estimators of all appropriate parameters from the normal equations XXB = X'Y Note: Do not write the matrices for a general factorial design model. Consider the particular factorial design described here.
3. Consider the two-factor model with interaction Suppose that there are a and b levels of the factors respectively. Now consider the set of equations (a) Show that the equations are not redundant. (b) Show that these equations are equivalent to the hypothesis of no interaction. (c) Thereby calculate the rank of the hypothesis of no interaction. (d) Show that the hypothesis is testable, provided there exists at least one sample from each combination of factor levels.
4. In a process development study on yield, four factors were studied, each at two levels: time (A), concentration (B), pressure (C) and temperature (D). A single replicate of a 24 design was run, and the resulting are shown in the Table below a. Construct a normal probability plot of the effect estimates. Which factors appear to have large effects? b. Conduct an analysis of variance using the normal probability plot in part (a) for guidance in forming an error...
Solve parts b and d by using SPSS
6.5 An engineer is interested in the effects of cutting speed (A), tool geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three replicates of a 2 factorial design are run. The results are as follows: Replicate Treatment B C Combination 22 31 32 43 35 34 50 55 47 46 44 40 37 36 60 50 54...
Problem #4 (25). In class. An engineer is interested in the effects of cutting speed (A), too! geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three replicates of a 23 factorial design are run. The results are as follows AİB Treatment combination Replicate 31 43 34 47 45 37 25 29 32 35 46 38 36 54 47 ас 41 abc B. Suppose that these...
1. An engineer is interested in the effects of cutting speed (A), tool geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three replicates of a 23 factorial design are run. The results are as follows: Treatment Replicate A B CCombinationI 22 31 32 43 35 34 55 47 44 45 40 37 60 50 54 39 41 25 29 50 46 38 36 ac bc...
Solve parts b and d by using SPSS
6.5 An engineer is interested in the effects of cutting speed (A), tool geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three replicates of a 2 factorial design are run. The results are as follows: Replicate Treatment B C Combination 22 31 32 43 35 34 50 55 47 46 44 40 37 36 60 50 54...
Problem #4 (25). In class. An engineer is interested in the effects of cutting speed (A), too! geometry (B), and cutting angle (C) on the life (in hours) of a machine tool. Two levels of each factor are chosen, and three replicates of a 23 factorial design are run. The results are as follows: Treatment combination Replicate 31 43 34 47 45 37 50 41 25 29 50 32 35 ab 40 ас bc abc 54 47 A. What is...
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