91Ó°ÊÓ

An experiment is conducted so that an engineer can gain insight into the influence of sealing temperature \(A\), cooling bar temperature \(B,\) percent polyethylene additive \(C,\) and pressure \(D\) on the seal strength (in grams per inch) of a. bread-wrapper Stock. A \(\frac{1}{2}\) fraction of a \(2^{\prime 1}\) factorial experiment is used with the defining contrast being \(A B C D .\) The data are: presented here. Perform an analysis of variance on main effects, and two-factor interactions, assuming that all three-factor and higher interactions are negligible. Use \(a=0.05\). $$ \begin{array}{rrrrr} A & B & C & D & \text { Response } \\ \hline-1 & -1^{*} & -1 & -1 & 6.6 \\ 1 & -1 & -1 & 1 & 6.9 \\ -1 & 1 & -1 & 1 & 7.9 \\ 1 & 1 & -1 & -1 & 6.1 \\ -1 & -1 & 1 & 1 & 9.2 \\ 1 & -1 & 1 & -1 & 6.8 \\ -1 & 1 & 1 & -1 & 10.4 \\ 1 & 1 & 1 & 1 & 7.3 \end{array} $$

Step by step solution

01

Calculation of Column Totals

First, calculate the total of each column and the total response. For example, the total of Column A is the sum of all values under Column A, and so on. Perform the same step for all columns, including the 'Response' column.

02

Calculation of Main Effects

Next, compute the main effects by subtracting the sum of the negative levels from the sum of positive levels, and dividing by 4 (the total number of observations at each level). Repeat this step for all main factors A, B, C, and D.

03

Calculation of Two-Factor Interactions

You calculate the interactions by multiplying the corresponding levels of the two factors for each observation, adding these products up, and dividing them by the total number of observations. For example, to find the AB interaction, you multiply the values in the ‘A’ and 'B' columns for each row, add these products, and then divide by 8 (the total number of observations). Repeat this process for all two-factor combinations: AB, AC, AD, BC, BD, and CD.

04

Analysis of Variance (ANOVA)

Now, perform an Analysis of Variance (ANOVA) with these main effects and interactions as inputs. ANOVA will yield a ‘sum of squares’ and a ‘mean square’ for every main effect and two-factor interaction, along with an F-value determined by dividing the ‘mean square’ by the ‘error mean square’. Then calculate the p-values.

05

Interpretation

Once the ANOVA is complete, determine which factors and interactions are statistically significant using a threshold of 0.05. Any factor or interaction having a p-value less than or equal to 0.05 is considered statistically significant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Analysis of Variance

Analysis of Variance (ANOVA) is a statistical method used to compare the means of three or more groups to determine if at least one of the group means is statistically different from the others. In the context of a factorial experiment, ANOVA decomposes the overall variability in the observed responses into components attributable to each factor and their interactions. These components are referred to as the 'sum of squares' for each factor. ANOVA calculates the 'mean square' by dividing these sums by their respective degrees of freedom. An F-value is then obtained by comparing these mean squares to the 'error mean square', which is the variability within the groups. A low p-value for the F-test indicates a high level of confidence that the corresponding factor or interaction has a significant effect on the response variable.

Factorial Design

Factorial design is a systematic way of investigating the effects of two or more experimental factors by varying them together instead of one at a time. In a factorial design, each level of a factor is tested in combination with each level of all other factors, leading to a comprehensive exploration of the experiment space. This type of design is powerful because it can identify not only the main effects of the individual factors but also any interaction effects between them. The fraction of a factorial design, such as the \( \frac{1}{2} \) fraction used in the given problem, uses only a subset of the possible factor combinations, which makes the experimentation process more efficient when dealing with a large number of factors.

Interaction Effects

Interaction effects occur when the effect of one experimental factor on the response variable depends on the level of another factor. In other words, it's the combined effect of two or more factors that is not simply additive. Detecting interaction effects is a key benefit of factorial design as they can reveal complex relationships between variables. For example, the AB interaction is investigated by looking at how the combined levels of factor A and B influence the response, beyond their individual main effects. When an interaction is statistically significant, it means the corresponding factors do not operate independently on the response and it is crucial to consider these relationships when interpreting the results of the experiment.

Statistical Significance

Statistical significance is a determination of whether the observed effects in an experiment are likely to be genuine or if they could have occurred by random chance. This is typically assessed using a p-value, which reflects the probability that the observed data would occur if there were no actual effect (null hypothesis). A commonly used threshold for statistical significance is \( p \leq 0.05 \), implying that there is less than a 5% chance that the observed effect is due to random variation alone. In the context of the experiment, if a main effect or interaction has a p-value at or below this threshold, it is considered statistically significant, meaning it is likely to have a real effect on the response variable.