In an experiment conducted at the Department of Mechanical...

In an experiment conducted at the Department of Mechanical Engineering and analyzed by the Statistics Consulting Center at the Virginia Polytechnic Institute and State University, a sensor detects an electrical charge each time a turbine blade makes one; rotation. The sensor then measures the amplitude of the electrical current. Six factors arc rprn $A$, temperature $B$, gap between blades $C$, gap between blade and casing $D$, location of input $E$, and location of detection $F .$ A $\frac{1}{4}$ fraction of a $2^{\circ}$ factorial experiment is used, with defining contrasts being $A B C E$ and $B C D F .$ The data are as follows: $$ \begin{array}{rrrrrrr} A & B & C & D & E & F & \text { Response } \\ \hline-1 & -1 & -1 & -1 & -1 & -1 & 3.89 \\ 1 & -1 & -1 & -\mathrm{i} & 1 & -1 & 10.46 \\ -1 & 1 & -1 & -1 & 1 & 1 & 25.98 \\ 1 & 1 & -1 & -1 & -1 & 1 & 39.88 \\ -1 & -1 & 1 & -1 & 1 & 1 & 61.88 \\ 1 & -1 & 1 & -1 & -1 & 1 & 3.22 \\ -1 & 1 & 1 & -1 & -1 & -1 & 8.94 \\ 1 & 1 & 1 & -1 & 1 & -1 & 20.29 \\ -1 & -1 & -1 & 1 & -1 & 1 & 32.07 \\ 1 & -1 & -1 & 1 & 1 & 1 & 50.76 \\ -1 & 1 & -1 & 1 & 1 & -1 & 2.80 \\ 1 & 1 & -1 & 1 & -1 & -1 & 8.15 \\ -1 & -1 & 1 & 1 & 1 & -1 & 16.80 \\ 1 & -1 & 1 & 1 & -1 & -1 & 25.47 \\ -1 & 1 & 1 & 1 & -1 & 1 & 44.44 \\ 1 & 1 & 1 & 1 & 1 & 1 & 2.45 \end{array} $$ Perform an analysis of variance on main effects, and two-factor interactions. assuming that all three-factor and higher interactions are negligible. Use $\alpha=0.05$.

In an experiment conducted at the Department of Mechanical Engineering and analyzed by the Statistics Consulting Center at the Virginia Polytechnic Institute and State University, a sensor detects an electrical charge each time a turbine blade makes one; rotation. The sensor then measures the amplitude of the electrical current. Six factors arc rprn $A$, temperature $B$, gap between blades $C$, gap between blade and casing $D$, location of input $E$, and location of detection $F .$ A $\frac{1}{4}$ fraction of a $2^{\circ}$ factorial experiment is used, with defining contrasts being $A B C E$ and $B C D F .$ The data are as follows:
$$
\begin{array}{rrrrrrr}
A & B & C & D & E & F & \text { Response } \\
\hline-1 & -1 & -1 & -1 & -1 & -1 & 3.89 \\
1 & -1 & -1 & -\mathrm{i} & 1 & -1 & 10.46 \\
-1 & 1 & -1 & -1 & 1 & 1 & 25.98 \\
1 & 1 & -1 & -1 & -1 & 1 & 39.88 \\
-1 & -1 & 1 & -1 & 1 & 1 & 61.88 \\
1 & -1 & 1 & -1 & -1 & 1 & 3.22 \\
-1 & 1 & 1 & -1 & -1 & -1 & 8.94 \\
1 & 1 & 1 & -1 & 1 & -1 & 20.29 \\
-1 & -1 & -1 & 1 & -1 & 1 & 32.07 \\
1 & -1 & -1 & 1 & 1 & 1 & 50.76 \\
-1 & 1 & -1 & 1 & 1 & -1 & 2.80 \\
1 & 1 & -1 & 1 & -1 & -1 & 8.15 \\
-1 & -1 & 1 & 1 & 1 & -1 & 16.80 \\
1 & -1 & 1 & 1 & -1 & -1 & 25.47 \\
-1 & 1 & 1 & 1 & -1 & 1 & 44.44 \\
1 & 1 & 1 & 1 & 1 & 1 & 2.45
\end{array}
$$
Perform an analysis of variance on main effects, and two-factor interactions. assuming that all three-factor and higher interactions are negligible. Use $\alpha=0.05$.

Key Concepts

Significance Testing (Alpha Level)

Significance testing in ANOVA involves comparing computed test statistics to critical values determined by a chosen significance level (commonly alpha = 0.05) to decide if an observed effect is statistically significant. This process helps determine whether the factor or interaction effects are unlikely to have occurred by chance, guiding decisions on which factors have meaningful impacts on the response variable.

Analysis of Variance (ANOVA)

ANOVA is a statistical method used to test for significant differences among group means and to assess the contributions of different sources of variation, such as main effects and interactions. In the context of designed experiments, ANOVA partitions the total variability in the response into components attributable to each factor and interaction, allowing for hypothesis testing using calculated p-values.

Two-Factor Interactions

Two-factor interactions occur when the effect of one factor on the response variable depends on the level of another factor. In the analysis of variance framework, these interactions are important to evaluate because they reveal whether factors work jointly in influencing the response, beyond their individual main effects.

Defining Contrasts and Aliasing Structure

In fractional factorial experiments, defining contrasts (or generators) establish the aliasing structure, which determines how main effects and interactions are confounded with each other. This aliasing occurs because not all possible factor combinations are run. Proper understanding of the defining contrasts helps in interpreting which effects are mixed together and guides the assumption that higher-order interactions (such as three-factor or higher interactions) can be neglected.

Fractional Factorial Design

This is an experimental design method that uses a fraction of the full factorial set of experiments to study the effects of several factors simultaneously. It is particularly useful when the number of factors is large, as it reduces the number of runs required. However, a key tradeoff is that some effects may be aliased with one another, meaning that the estimated effects represent combinations of factor influences rather than isolated contributions.

Main Effects

Main effects refer to the impact of a single factor on the response variable, averaged over the levels of all other factors. In the context of an analysis of variance (ANOVA), main effects are evaluated to determine if changes in the factor levels produce significant differences in the outcome measured in the experiment.

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