Question

Problem 2 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 2 design was run and the resulting data are shown in the following table Yield (Ibs) 67 128 86 134 97 92 132 128 81 159 114 163 93 109 122 140 Actual Run Order 9 12 13 14 15 10 16 The factor level settings are as follows Factor A (hr) B (%) C (psi) D (F) Low (- High (+) 12 40 200 16 80 300 (a) Estimate the factor effects by hand (you may use excel). Which effects appear to be large? (b) Perform an analysis of variance by hand to confirm your conclusions for part (a). What terms are significant at a-0.05? (c) Confirm your results in Minitab. Construct a normal probability plot of the factor effects, perform an ANOVA and analyze the ANOVA residuals. (d) Construct a regression model by hand for predicting yield. Use the significant terms you identified in your part (b) ANOVA (e) Confirm you part (d) results using Minitab. Analyze the residuals from your regression analysis (f) Based on an analysis of significant main effect and interaction plots, what levels of A, B, C, and D would you recommend using to maximize the yield? (g) Using the regression model found in part (e), predict the yield at 2.5 h, 15 % 65 psi and a temperature of 250 degrees

Answer 1

Here we have given four factors each at two levels. The design is called 2⁴ design.

The four factors are :

A : Time

B : Concentration

C : Pressure

D : Temperature.

First we have to construct 2⁴ factorial design.

This we can done in MINITAB.

steps :

STAT --> DOE --> Factorial --> Create factorial design --> Type of design : 2-level factorial (default generators) --> Number of factors : 4 --> Designs --> Click on full factorial --> Number of center points per block : 0 --> Number of replicates for corner points : 1 --> Number of blocks : 1 --> ok --> Options : Store design in worksheet --> ok --> Results --> Printed results : Select last option --> ok --> ok

————— 7/6/2017 9:40:02 AM ————————————————————

Results for: Worksheet 4

Full Factorial Design

Factors: 4 Base Design: 4, 16
Runs: 16 Replicates: 1
Blocks: 1 Center pts (total): 0

All terms are free from aliasing.

Design Table

Run A B C D
1 - - - -
2 + - - -
3 - + - -
4 + + - -
5 - - + -
6 + - + -
7 - + + -
8 + + + -
9 - - - +
10 + - - +
11 - + - +
12 + + - +
13 - - + +
14 + - + +
15 - + + +
16 + + + +

Now we have to analyze this design.

steps :

ENTER data into MINITAB sheet --> Stat --> DOE --> Factorial --> ANalyze factorial design --> Responses : Select data column --> Terms : select --> ok --> Results --> Display of results: select 4th option --> Display of least squares means --> Selected terms : Select four factors --> ok --> ok

————— 7/6/2017 9:40:02 AM ————————————————————

Factorial Fit: observation versus A, B, C, D

Estimated Effects and Coefficients for observation (coded units)

Term Effect Coef SE Coef T P
Constant 37.875 0.7500 50.50 0.013
A 10.000 5.000 0.7500 6.67 0.095
B 0.750 0.375 0.7500 0.50 0.705
C 4.250 2.125 0.7500 2.83 0.216
D 6.750 3.375 0.7500 4.50 0.139
A*B -2.500 -1.250 0.7500 -1.67 0.344
A*C -9.000 -4.500 0.7500 -6.00 0.105
A*D 9.000 4.500 0.7500 6.00 0.105
B*C 0.250 0.125 0.7500 0.17 0.895
B*D -0.250 -0.125 0.7500 -0.17 0.895
C*D 0.250 0.125 0.7500 0.17 0.895
A*B*C 2.500 1.250 0.7500 1.67 0.344
A*B*D 1.500 0.750 0.7500 1.00 0.500
A*C*D -1.000 -0.500 0.7500 -0.67 0.626
B*C*D -1.750 -0.875 0.7500 -1.17 0.451

S = 3 R-Sq = 99.35% R-Sq(adj) = 90.29%

Analysis of Variance for observation (coded units)

Source DF Seq SS Adj SS Adj MS F P
Main Effects 4 656.75 656.750 164.187 18.24 0.174
2-Way Interactions 6 673.75 673.750 112.292 12.48 0.213
3-Way Interactions 4 50.25 50.250 12.562 1.40 0.555
Residual Error 1 9.00 9.000 9.000
Total 15 1389.75

Obs StdOrder observation Fit SE Fit Residual St Resid
1 1 26.0000 25.2500 2.9047 0.7500 1.00
2 2 40.0000 40.7500 2.9047 -0.7500 -1.00
3 3 30.0000 30.7500 2.9047 -0.7500 -1.00
4 4 34.0000 33.2500 2.9047 0.7500 1.00
5 5 37.0000 37.7500 2.9047 -0.7500 -1.00
6 6 33.0000 32.2500 2.9047 0.7500 1.00
7 7 43.0000 42.2500 2.9047 0.7500 1.00
8 8 33.0000 33.7500 2.9047 -0.7500 -1.00
9 9 21.0000 21.7500 2.9047 -0.7500 -1.00
10 10 55.0000 54.2500 2.9047 0.7500 1.00
11 11 28.0000 27.2500 2.9047 0.7500 1.00
12 12 52.0000 52.7500 2.9047 -0.7500 -1.00
13 13 41.0000 40.2500 2.9047 0.7500 1.00
14 14 47.0000 47.7500 2.9047 -0.7500 -1.00
15 15 37.0000 37.7500 2.9047 -0.7500 -1.00
16 16 49.0000 48.2500 2.9047 0.7500 1.00

Least Squares Means for observation

Mean SE Mean
A
-1 32.88 1.061
1 42.88 1.061
B
-1 37.50 1.061
1 38.25 1.061
C
-1 35.75 1.061
1 40.00 1.061
D
-1 34.50 1.061
1 41.25 1.061
A*C
-1 -1 26.25 1.500
1 -1 45.25 1.500
-1 1 39.50 1.500
1 1 40.50 1.500
A*D
-1 -1 34.00 1.500
1 -1 35.00 1.500
-1 1 31.75 1.500
1 1 50.75 1.500
B*C
-1 -1 35.50 1.500
1 -1 36.00 1.500
-1 1 39.50 1.500
1 1 40.50 1.500
B*D
-1 -1 34.00 1.500
1 -1 35.00 1.500
-1 1 41.00 1.500
1 1 41.50 1.500
C*D
-1 -1 32.50 1.500
1 -1 36.50 1.500
-1 1 39.00 1.500
1 1 43.50 1.500
A*B*C
-1 -1 -1 23.50 2.121
1 -1 -1 47.50 2.121
-1 1 -1 29.00 2.121
1 1 -1 43.00 2.121
-1 -1 1 39.00 2.121
1 -1 1 40.00 2.121
-1 1 1 40.00 2.121
1 1 1 41.00 2.121
A*B*D
-1 -1 -1 31.50 2.121
1 -1 -1 36.50 2.121
-1 1 -1 36.50 2.121
1 1 -1 33.50 2.121
-1 -1 1 31.00 2.121
1 -1 1 51.00 2.121
-1 1 1 32.50 2.121
1 1 1 50.50 2.121
A*C*D
-1 -1 -1 28.00 2.121
1 -1 -1 37.00 2.121
-1 1 -1 40.00 2.121
1 1 -1 33.00 2.121
-1 -1 1 24.50 2.121
1 -1 1 53.50 2.121
-1 1 1 39.00 2.121
1 1 1 48.00 2.121
B*C*D
-1 -1 -1 33.00 2.121
1 -1 -1 32.00 2.121
-1 1 -1 35.00 2.121
1 1 -1 38.00 2.121
-1 -1 1 38.00 2.121
1 -1 1 40.00 2.121
-1 1 1 44.00 2.121
1 1 1 43.00 2.121

Alias Structure
I
A
B
C
D
A*B
A*C
A*D
B*C
B*D
C*D
A*B*C
A*B*D
A*C*D
B*C*D

Overall significance :

Here test statistic follows F-distribution.

Here overall significance for the term main effects, 2-way interaction and 3-way interaction.

We see that all the three factors, 2-way interactions and 3-way interactions are insignificant.

Also we can see that all the terms are insignificant at 5% level of significance.

Here we can see from the normal probability plot the data follows normal distribution.