Questions and Answers
What are carotenoids?
What is the well-known carotenoid mentioned in the text?
Beta-carotene
Astaxanthin is a weak antioxidant.
False
In the example used, a surprisingly small set of ___ runs was utilized.
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What is the optimization experiment in the given scenario called, where two factors are run at two levels each near the current operating conditions?
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What are the main effects in a factorial experiment?
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In factorial experiments, factors are varied independently of each other.
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Factorial experiments involve investigating all possible combinations of the levels of the factors in each __________.
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In Table 14-1, what is the AB interaction effect?
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What is illustrated by Figure 14-3 in terms of the interaction between A and B?
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Factorial experiments are the only way to discover interactions between variables. (True/False)
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What is a key disadvantage of using the one-factor-at-a-time design approach?
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What is the simplest type of factorial experiment that involves only two factors?
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What does the ANOVA stand for?
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Hypothesis H0 in a two-factor factorial experiment states that there is no main effect of factor A (True/False).
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The total sum of squares in a two-factor factorial experiment is measured by ________.
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What is the conclusion drawn regarding the main effects of primer type and application method on adhesion force based on the given ANOVA results?
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What is the main takeaway from the graph of average adhesion force versus primer types for both application methods shown in Figure 14-10?
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According to the analysis results, Primer Type 2 is the most effective primer type for adhesion force.
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What type of design is discussed in the provided content?
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Which factor has a main effect of feed in the analysis of variance table?
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How many degrees of freedom are associated with Factor C?
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What are the mean squares for factor A, B, interaction, and error in a two-factor factorial experiment?
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What is the formula for the F test used to test the effect of factor A in a factorial experiment?
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What degrees of freedom are used in the F test for factor B in a factorial experiment?
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In a factorial experiment, if the null hypothesis about interaction effects being zero is true, the mean squares are unbiased estimates.
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Write down a model for the experiment described in the article from the Journal of Testing and Evaluation.
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Which statistical method is mentioned to isolate the effects of anneal conditions on base current?
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In a factorial experiment, there must be at least two replicates to compute an error sum of squares.
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The response variable in the experiment described by Johnson and Leone is __________ growth rate.
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State the hypotheses of interest in the first experiment described in section 14-1.
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What is one possible assumption made in the case of having one observation per cell in a two-factor factorial experiment?
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State the hypotheses of interest in the second experiment described in section 14-2.
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What is being investigated in the experiment involving firing temperature and furnace position as described in section 14-4?
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What does the article in Technometrics describe?
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What are the four insulation types tested and at how many different temperatures in the experiment described in section 14-5?
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In the experiment described in section 14-7, both polysilicon doping level and anneal conditions affect base current.
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Study Notes
Design of Experiments with Several Factors
- Carotenoids are fat-soluble pigments found in fruits and vegetables, and an experiment was conducted to study the production of astaxanthin, a strong antioxidant, considering seven variables: photon flux density, and concentrations of nitrogen, phosphorous, magnesium, acetate, ferrous, and NaCl.
- The experiment involved studying the effects of these factors and their combinations, but using a fraction of the full set of tests to provide important information about the effects of these variables.
- This chapter focuses on the design and analysis of experiments involving several factors, which are widely used in modern engineering development and scientific studies.
Importance of Experimental Design
- Experiments are performed in all engineering and scientific disciplines to learn about how systems and processes work.
- The validity of conclusions drawn from an experiment depends on how the experiment was conducted, making experimental design a crucial aspect.
- Experimental design is particularly important in improving the performance of manufacturing processes, developing new processes, and designing new products.
Factorial Experiments
- A factorial experiment is a powerful technique for studying the effects of multiple factors, where all possible combinations of factor levels are investigated.
- The effect of a factor is defined as the change in response produced by a change in the level of the factor, and is called a main effect.
- Factorial experiments can be used to estimate the magnitude and direction of factor effects, and to identify the critical process variables that affect the response.
Examples of Experimental Design
- Process Characterization Experiment: An experiment was conducted to study the effects of controllable and uncontrollable factors on the SMT process, which involves assembling electronic components to a printed circuit board.
- Optimization Experiment: An experiment was conducted to determine the region in the important factors that leads to an optimal response, such as minimum cost or maximum yield.
- Product Design Example: An experiment was conducted to design a door hinge for an automobile, considering factors such as roller travel distance, spring height, and horizontal distance from pivot to spring.
Analysis of Factorial Experiments
- The analysis of variance (ANOVA) is used as a tool for statistical data analysis in factorial experiments.
- Graphical methods are also useful in analyzing the data from designed experiments.### Main Effects of Factors
- The main effect of a factor is the difference between the average response at the high level and the average response at the low level of the factor.
- Example: In a factorial experiment, the main effect of factor A is the difference between the average response at the high level of A and the average response at the low level of A.
Interaction Between Factors
- Interaction occurs when the difference in response between the levels of one factor is not the same at all levels of the other factor.
- Example: In a factorial experiment, the effect of factor A depends on the level chosen for factor B, indicating an interaction between A and B.
- When interaction is present, the main effects of the factors involved may not have much practical meaning.
Estimating Interaction Effects
- In a factorial experiment, the interaction effect is the difference in the diagonal averages.
- Example: In a 2x2 factorial experiment, the interaction effect is one-half the difference between the A effects at the two levels of B.
Graphical Representation of Interaction
- Two-factor interaction plots can be used to illustrate the interaction between factors.
- Parallel lines on the graph indicate no interaction between factors, while non-parallel lines indicate interaction.
Factorial Experiments
- Factorial experiments are the only way to discover interactions between variables.
- A factorial experiment involves varying multiple factors simultaneously, rather than one at a time.
Limitations of One-Factor-at-a-Time Approach
- The one-factor-at-a-time approach cannot detect interactions between variables.
- This approach is inefficient and may require more experimentation than a factorial experiment.
Two-Factor Factorial Experiments
- A two-factor factorial experiment involves two factors, each with multiple levels.
- The experiment is a completely randomized design, with each replicate containing all treatment combinations.
- The observations can be described by a linear statistical model.
Analysis of Variance (ANOVA)
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ANOVA is used to test the hypotheses of no main effect for each factor and no interaction effect.
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The two-way analysis of variance is used to test these hypotheses in a two-factor factorial experiment.### Two-Factor Factorial Experiments
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A two-factor factorial experiment involves two factors, A and B, with multiple levels, and replication.
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The effects of the factors are specifically chosen by the experimenter, and inferences are confined to these levels only.
Notation
- yi.. denotes the total of the observations taken at the ith level of factor A.
- y.j. denotes the total of the observations taken at the jth level of factor B.
- yij. denotes the total of the observations in the ijth cell.
- y... denotes the grand total of all the observations.
- Corresponding row, column, cell, and grand averages are denoted by yi.., y.j., yij., and y... respectively.
Hypotheses
- Three hypotheses are tested:
- H0: 1 = 2 = … = a = 0 (no main effect of factor A)
- H0: 1 = 2 = … = b = 0 (no main effect of factor B)
- H0: ()11 = ()12 = … = ()ab = 0 (no interaction)
Analysis of Variance (ANOVA)
- Total sum of squares (SST) is decomposed into component parts:
- Sum of squares for factor A (SSA)
- Sum of squares for factor B (SSB)
- Sum of squares for interaction between A and B (SSAB)
- Error sum of squares (SSE)
- SST = SSA + SSB + SSAB + SSE
- Degrees of freedom: abn - 1 = a - 1 + b - 1 + (a - 1)(b - 1) + ab(n - 1)
Mean Squares and F-Ratios
- Mean squares are calculated by dividing each sum of squares by the corresponding number of degrees of freedom:
- MSA = SSA / (a - 1)
- MSB = SSB / (b - 1)
- MSAB = SSAB / ((a - 1)(b - 1))
- MSE = SSE / ab(n - 1)
- F-ratios are calculated by dividing the mean squares for each factor and interaction by the mean square error:
- F0 = MSA / MSE
- F0 = MSB / MSE
- F0 = MSAB / MSE
Expected Values of Mean Squares
- Assuming factors A and B are fixed, the expected values of the mean squares are:
- E(MSA) = 2 + bn(g2i / a)
- E(MSB) = 2 + an(g2j / b)
- E(MSAB) = 2 + n(g2ij / ab)
- E(MSE) = 2
F-Tests
- F-tests are used to determine whether the main effects and interaction are significant.
- The null hypotheses are rejected if the F-ratio exceeds the critical value or if the P-value is less than the chosen significance level.
Example: Aircraft Primer Paint
- A two-factor factorial experiment was conducted to investigate the effect of paint primer type and application method on paint adhesion.
- The data were analyzed using ANOVA, and the results indicated that the main effects of primer type and application method affect adhesion force, but there is no indication of interaction between these factors.
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Description
Test your knowledge of carotenoids, a specific type of antioxidant, and factorial experiments in chemistry. This quiz covers the basics of carotenoids and experimental design.
