Question

In an anticle in Quality Engineering ("An Application of Fractional Factorial Experimental Designs," 1988, Vot. L. pp. 19-23). M. B. Kilgo describes an experiment to determine the effect of $\mathrm{CO}_2$ pressure (A). $\mathrm{CO}_2$ temperature ( $B$ ), peanut TABLE P8.6 can't copy Design and Results for Wine Tasting Experiment TABLE P8.7 Factor Levels for the Experiment in Problem 8.28 $$ \begin{array}{cccccc} \text { Coded Level } & \begin{array}{c} \text { A, } \\ \text { Pressure } \\ \text { (bar) } \end{array} & \begin{array}{c} \text { B, } \\ \text { Temp, } \\ \text { ( }{ }^{\circ} \text { ) } \end{array} & \begin{array}{c} \text { C, Moisture } \\ \text { (\% by weight) } \end{array} & \begin{array}{c} \text { D, Flow } \\ \text { (iters/min) } \end{array} & \begin{array}{c} \text { Part. Size } \\ \text { (mm) } \end{array} \\ \hline-1 & 415 & 25 & 5 & 40 & 1.28 \\ 1 & 550 & 95 & 15 & 60 & 4.05 \end{array} $$ moisture (C) $\mathrm{CO}_2$ flow rate (D), and peanut particle size (E) on the total yield of oil per batch of peanuts (y). The levels that she used for these factors are shown in Table P8.7. She conducted the 16 -run fractional factorial experiment shown in Table P8.8. TABLE P8.8 The Peanut Oil Experiment $$ \begin{array}{lrrrrl} \boldsymbol{A} & \boldsymbol{B} & \boldsymbol{C} & \boldsymbol{D} & \boldsymbol{E} & \boldsymbol{y} \\ \hline 415 & 25 & 5 & 40 & 1.28 & 63 \\ 550 & 25 & 5 & 40 & 4.05 & 21 \\ 415 & 95 & 5 & 40 & 4.05 & 36 \\ 550 & 95 & 5 & 40 & 1.28 & 99 \\ 415 & 25 & 15 & 40 & 4.05 & 24 \\ 550 & 25 & 15 & 40 & 1.28 & 66 \\ 415 & 95 & 15 & 40 & 1.28 & 71 \\ 550 & 95 & 15 & 40 & 4.05 & 54 \\ 415 & 25 & 5 & 60 & 4.05 & 23 \\ 550 & 25 & 5 & 60 & 1.28 & 74 \\ 415 & 95 & 5 & 60 & 1.28 & 80 \\ 550 & 95 & 5 & 60 & 4.05 & 33 \\ 415 & 25 & 15 & 60 & 1.28 & 63 \\ 550 & 25 & 15 & 60 & 4.05 & 21 \\ 415 & 95 & 15 & 60 & 4.05 & 44 \\ 550 & 95 & 15 & 60 & 1.28 & 96 \end{array} $$ (a) What type of design has been used? Identify the defining relation and the alias relationships. (b) Estimate the factor effects and use a normal probability plot to tentatively identify the important factors. (c) Perform an appropriate statistical analysis to test the hypotheses that the factors identified in part (b) above have a significant effect on the yield of peanut oil. (d) Fit a model that could be used to predict peanut oil yield in terms of the factors that you have identified as important. (e) Analyze the residuals from this experiment and comment on model adequacy. 

   In an anticle in Quality Engineering ("An Application of Fractional Factorial Experimental Designs," 1988, Vot. L. pp. 19-23). M. B. Kilgo describes an experiment to determine the effect of $\mathrm{CO}_2$ pressure (A). $\mathrm{CO}_2$ temperature ( $B$ ), peanut
TABLE P8.6 can't copy
Design and Results for Wine Tasting Experiment
TABLE P8.7
Factor Levels for the Experiment in Problem 8.28
$$
\begin{array}{cccccc}
\text { Coded Level } & \begin{array}{c}
\text { A, } \\
\text { Pressure } \\
\text { (bar) }
\end{array} & \begin{array}{c}
\text { B, } \\
\text { Temp, } \\
\text { ( }{ }^{\circ} \text { ) }
\end{array} & \begin{array}{c}
\text { C, Moisture } \\
\text { (\% by weight) }
\end{array} & \begin{array}{c}
\text { D, Flow } \\
\text { (iters/min) }
\end{array} & \begin{array}{c}
\text { Part. Size } \\
\text { (mm) }
\end{array} \\
\hline-1 & 415 & 25 & 5 & 40 & 1.28 \\
1 & 550 & 95 & 15 & 60 & 4.05
\end{array}
$$
moisture (C) $\mathrm{CO}_2$ flow rate (D), and peanut particle size (E) on the total yield of oil per batch of peanuts (y). The levels that she used for these factors are shown in Table P8.7. She conducted the 16 -run fractional factorial experiment shown in Table P8.8.
TABLE P8.8
The Peanut Oil Experiment
$$
\begin{array}{lrrrrl}
\boldsymbol{A} & \boldsymbol{B} & \boldsymbol{C} & \boldsymbol{D} & \boldsymbol{E} & \boldsymbol{y} \\
\hline 415 & 25 & 5 & 40 & 1.28 & 63 \\
550 & 25 & 5 & 40 & 4.05 & 21 \\
415 & 95 & 5 & 40 & 4.05 & 36 \\
550 & 95 & 5 & 40 & 1.28 & 99 \\
415 & 25 & 15 & 40 & 4.05 & 24 \\
550 & 25 & 15 & 40 & 1.28 & 66 \\
415 & 95 & 15 & 40 & 1.28 & 71 \\
550 & 95 & 15 & 40 & 4.05 & 54 \\
415 & 25 & 5 & 60 & 4.05 & 23 \\
550 & 25 & 5 & 60 & 1.28 & 74 \\
415 & 95 & 5 & 60 & 1.28 & 80 \\
550 & 95 & 5 & 60 & 4.05 & 33 \\
415 & 25 & 15 & 60 & 1.28 & 63 \\
550 & 25 & 15 & 60 & 4.05 & 21 \\
415 & 95 & 15 & 60 & 4.05 & 44 \\
550 & 95 & 15 & 60 & 1.28 & 96
\end{array}
$$
(a) What type of design has been used? Identify the defining relation and the alias relationships.
(b) Estimate the factor effects and use a normal probability plot to tentatively identify the important factors.
(c) Perform an appropriate statistical analysis to test the hypotheses that the factors identified in part (b) above have a significant effect on the yield of peanut oil.
(d) Fit a model that could be used to predict peanut oil yield in terms of the factors that you have identified as important.
(e) Analyze the residuals from this experiment and comment on model adequacy.
Show more…
Design and Analysis of Experiments
Design and Analysis of Experiments
Douglas C.… 7th Edition
Chapter 8, Problem 28 ↓

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The experiment described is a fractional factorial design. This type of design is used when the number of factors is large, and a full factorial design would require too many runs. In this case, a 2^5-1 fractional factorial design is used, as there are 5 factors  Show more…

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In an anticle in Quality Engineering ("An Application of Fractional Factorial Experimental Designs," 1988, Vot. L. pp. 19-23). M. B. Kilgo describes an experiment to determine the effect of $\mathrm{CO}_2$ pressure (A). $\mathrm{CO}_2$ temperature ( $B$ ), peanut TABLE P8.6 can't copy Design and Results for Wine Tasting Experiment TABLE P8.7 Factor Levels for the Experiment in Problem 8.28 $$ \begin{array}{cccccc} \text { Coded Level } & \begin{array}{c} \text { A, } \\ \text { Pressure } \\ \text { (bar) } \end{array} & \begin{array}{c} \text { B, } \\ \text { Temp, } \\ \text { ( }{ }^{\circ} \text { ) } \end{array} & \begin{array}{c} \text { C, Moisture } \\ \text { (\% by weight) } \end{array} & \begin{array}{c} \text { D, Flow } \\ \text { (iters/min) } \end{array} & \begin{array}{c} \text { Part. Size } \\ \text { (mm) } \end{array} \\ \hline-1 & 415 & 25 & 5 & 40 & 1.28 \\ 1 & 550 & 95 & 15 & 60 & 4.05 \end{array} $$ moisture (C) $\mathrm{CO}_2$ flow rate (D), and peanut particle size (E) on the total yield of oil per batch of peanuts (y). The levels that she used for these factors are shown in Table P8.7. She conducted the 16 -run fractional factorial experiment shown in Table P8.8. TABLE P8.8 The Peanut Oil Experiment $$ \begin{array}{lrrrrl} \boldsymbol{A} & \boldsymbol{B} & \boldsymbol{C} & \boldsymbol{D} & \boldsymbol{E} & \boldsymbol{y} \\ \hline 415 & 25 & 5 & 40 & 1.28 & 63 \\ 550 & 25 & 5 & 40 & 4.05 & 21 \\ 415 & 95 & 5 & 40 & 4.05 & 36 \\ 550 & 95 & 5 & 40 & 1.28 & 99 \\ 415 & 25 & 15 & 40 & 4.05 & 24 \\ 550 & 25 & 15 & 40 & 1.28 & 66 \\ 415 & 95 & 15 & 40 & 1.28 & 71 \\ 550 & 95 & 15 & 40 & 4.05 & 54 \\ 415 & 25 & 5 & 60 & 4.05 & 23 \\ 550 & 25 & 5 & 60 & 1.28 & 74 \\ 415 & 95 & 5 & 60 & 1.28 & 80 \\ 550 & 95 & 5 & 60 & 4.05 & 33 \\ 415 & 25 & 15 & 60 & 1.28 & 63 \\ 550 & 25 & 15 & 60 & 4.05 & 21 \\ 415 & 95 & 15 & 60 & 4.05 & 44 \\ 550 & 95 & 15 & 60 & 1.28 & 96 \end{array} $$ (a) What type of design has been used? Identify the defining relation and the alias relationships. (b) Estimate the factor effects and use a normal probability plot to tentatively identify the important factors. (c) Perform an appropriate statistical analysis to test the hypotheses that the factors identified in part (b) above have a significant effect on the yield of peanut oil. (d) Fit a model that could be used to predict peanut oil yield in terms of the factors that you have identified as important. (e) Analyze the residuals from this experiment and comment on model adequacy.
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Key Concepts

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Residual Analysis
Residual analysis is a method for assessing the adequacy of a fitted model by examining the discrepancies between observed and predicted values. This analysis helps to check assumptions such as normality, constant variance, and independence of errors, thereby validating whether the model is appropriate for the data.
Fractional Factorial Design
This concept involves selecting a subset of all possible experimental runs in a factorial experiment to efficiently estimate primary effects with a reduced number of experiments. Fractional factorial designs are particularly useful when full factorial experiments would be too costly or time-consuming, and they involve deliberate confounding of interactions with main effects in a controlled manner.
Defining Relation and Aliasing Structure
A defining relation in fractional factorial designs specifies the pattern of confounding that inherently exists in the experimental runs, outlining which effects are aliased with each other. Understanding the alias structure is critical for identifying which effects can be uniquely estimated and which may be confounded, ultimately helping in interpreting the experiment’s outcomes correctly.
Effect Estimation
Effect estimation refers to the process of quantifying the influence of each factor (or combination of factors) on the response variable. It involves computing the difference in the response when factors are at high and low levels, providing an assessment of which factors have a substantial impact.
Normal Probability Plot
A normal probability plot is a diagnostic tool used to assess whether a set of data, such as estimated effects, follows a normal distribution. In the context of experimental design, it is used to visually identify which factor effects deviate from normality, thereby suggesting that those deviations are likely due to real effects rather than random noise.
Statistical Hypothesis Testing
Statistical hypothesis testing in the context of designed experiments is used to determine whether the observed effects of factors are statistically significant. By comparing test statistics to critical values or using p-values, researchers decide whether to reject the null hypothesis that a given factor has no effect on the response.
Regression Modeling
Regression modeling in experimental design involves developing a mathematical model that relates the response variable to one or more predictor variables (factors). This model is used not only for understanding the effects of the factors but also for predicting future outcomes based on different factor settings.
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Problem 8-49 from Design and Analysis of Experiments (Montgomery, 8th Edition): In an article in Quality Engineering ("An Application of Fractional Factorial Experimental Designs," 1988, Vol. 1 pp. 19-23), M.B. Kilgo describes an experiment to determine the effect of CO2 pressure (A), CO2 temperature (B), peanut moisture (C), CO2 flow rate (D), and peanut particle size (E) on the total yield of oil per batch of peanuts (y). The levels she used for these factors are as follows: Coded Level A B C D E Pressure Temp Moisture Flow Particle Size (bar) (C) (% by weight) (liters/min) (mm) -1 415 25 5 40 1.28 1 550 95 15 60 4.05 She conducted the 16-run fractional factorial experiment shown below: A B C D E y 1 415 25 5 40 1.28 63 2 550 25 5 40 4.05 21 3 415 95 5 40 4.05 36 4 550 95 5 40 1.28 99 5 415 25 15 40 4.05 24 6 550 25 15 40 1.28 66 7 415 95 15 40 1.28 71 8 550 95 15 40 4.05 54 9 415 25 5 60 4.05 23 10 550 25 5 60 1.28 74 11 415 95 5 60 1.28 80 12 550 95 5 60 4.05 33 13 415 25 15 60 1.28 63 14 550 25 15 60 4.05 21 15 415 95 15 60 4.05 44 16 550 95 15 60 1.28 96 a.) What type of design has been used? Identify the defining relation and the alias relationships. b.) By Minitab, estimate the factor effects and use a normal probability plot to tentatively identify the important factors. c.) Perform an appropriate statistical analysis to test the hypothesis that the factors identified in part above have a significant effect on the yield of peanut oil. d.) Fit a model that could be used to predict peanut oil yield in terms of the factors that you have identified as important. e.) Analyze the residuals from this experiment and comment on model adequacy.
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