A tasty carbonated beverage is being bottled in a factory, and the manufacturer
ID: 3208871 • Letter: A
Question
A tasty carbonated beverage is being bottled in a factory, and the manufacturer is interested in reducing the deviation in fill height within the bottle. Three factors are suspected of causing the deviation in fill height and a 2' experiment was conducted in which the factors and levels were: The experiment was conducted and the fill height deviation was recorded as: Calculate the main effects and interactions. Fit a linear regression model to these data of the form y = beta_0 + beta_1 x_1 + beta_2 x_2 + beta_3 x_3 + beta_12 x_1 x_2 + beta_13 x_1 x_3 + beta_25 x_2 x_3 beta_125 x_1 x_2 x_3 using coded variables. Include all effects and interactions in the model, even if they are small. Rewrite the model using natural variables. Include all effects and interactions in the model, even if they are small. Produce a residual plot for the model. What does the plot tell you? Use the model to construct three contour plots to assist in interpreting the results of the experiment. B vs A C vs A and C vs B.Explanation / Answer
a]
Using Minitab
Full Factorial Design
Factors: 3 Base Design: 3, 8
Runs: 8 Replicates: 1
Blocks: 1 Center pts (total): 0
Factorial Fit: R1, R2, R3, R4
Factorial Fit: R1 versus A, B, C
Estimated Effects and Coefficients for R1 (coded units)
Term Effect Coef
Constant 0.750
A -1.000 -0.500
B -3.000 -1.500
C 2.000 1.000
A*B 1.500 0.750
A*C -0.500 -0.250
B*C 0.500 0.250
A*B*C 3.000 1.500
Analysis of Variance for R1 (coded units)
Source DF Seq SS Adj SS Adj MS F P
Main Effects 3 28.0000 28.0000 9.3333 * *
A 1 2.0000 2.0000 2.0000 * *
B 1 18.0000 18.0000 18.0000 * *
C 1 8.0000 8.0000 8.0000 * *
2-Way Interactions 3 5.5000 5.5000 1.8333 * *
A*B 1 4.5000 4.5000 4.5000 * *
A*C 1 0.5000 0.5000 0.5000 * *
B*C 1 0.5000 0.5000 0.5000 * *
3-Way Interactions 1 18.0000 18.0000 18.0000 * *
A*B*C 1 18.0000 18.0000 18.0000 * *
Residual Error 0 * * *
Total 7 51.5000
Factorial Fit: R2 versus A, B, C
Estimated Effects and Coefficients for R2 (coded units)
Term Effect Coef
Constant 1.250
A -0.500 -0.250
B -2.500 -1.250
C 1.000 0.500
A*B 0.500 0.250
A*C -1.000 -0.500
B*C 0.000 0.000
A*B*C 2.000 1.000
Analysis of Variance for R2 (coded units)
Source DF Seq SS Adj SS Adj MS F P
Main Effects 3 15.0000 15.0000 5.0000 * *
A 1 0.5000 0.5000 0.5000 * *
B 1 12.5000 12.5000 12.5000 * *
C 1 2.0000 2.0000 2.0000 * *
2-Way Interactions 3 2.5000 2.5000 0.8333 * *
A*B 1 0.5000 0.5000 0.5000 * *
A*C 1 2.0000 2.0000 2.0000 * *
B*C 1 0.0000 0.0000 0.0000 * *
3-Way Interactions 1 8.0000 8.0000 8.0000 * *
A*B*C 1 8.0000 8.0000 8.0000 * *
Residual Error 0 * * *
Total 7 25.5000
Factorial Fit: R3 versus A, B, C
Estimated Effects and Coefficients for R3 (coded units)
Term Effect Coef
Constant 0.7500
A 0.0000 0.0000
B -1.5000 -0.7500
C 1.5000 0.7500
A*B 1.0000 0.5000
A*C -1.0000 -0.5000
B*C 0.5000 0.2500
A*B*C 2.0000 1.0000
Analysis of Variance for R3 (coded units)
Source DF Seq SS Adj SS Adj MS F P
Main Effects 3 9.0000 9.00000 3.00000 * *
A 1 0.0000 0.00000 0.00000 * *
B 1 4.5000 4.50000 4.50000 * *
C 1 4.5000 4.50000 4.50000 * *
2-Way Interactions 3 4.5000 4.50000 1.50000 * *
A*B 1 2.0000 2.00000 2.00000 * *
A*C 1 2.0000 2.00000 2.00000 * *
B*C 1 0.5000 0.50000 0.50000 * *
3-Way Interactions 1 8.0000 8.00000 8.00000 * *
A*B*C 1 8.0000 8.00000 8.00000 * *
Residual Error 0 * * *
Total 7 21.5000
Factorial Fit: R4 versus A, B, C
Estimated Effects and Coefficients for R4 (coded units)
Term Effect Coef
Constant 1.250
A 0.500 0.250
B -3.000 -1.500
C 2.500 1.250
A*B 1.000 0.500
A*C -0.500 -0.250
B*C 0.000 0.000
A*B*C 3.000 1.500
Analysis of Variance for R4 (coded units)
Source DF Seq SS Adj SS Adj MS F P
Main Effects 3 31.0000 31.0000 10.3333 * *
A 1 0.5000 0.5000 0.5000 * *
B 1 18.0000 18.0000 18.0000 * *
C 1 12.5000 12.5000 12.5000 * *
2-Way Interactions 3 2.5000 2.5000 0.8333 * *
A*B 1 2.0000 2.0000 2.0000 * *
A*C 1 0.5000 0.5000 0.5000 * *
B*C 1 0.0000 0.0000 0.0000 * *
3-Way Interactions 1 18.0000 18.0000 18.0000 * *
A*B*C 1 18.0000 18.0000 18.0000 * *
Residual Error 0 * * *
Total 7 51.5000
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