Factor B Level 1 Level 2 Level 3 104 111 95 82 104

step 1 Okay, so for A, we're going to determine whether there is a significant interaction between fact

Factor B Level 1 Level 2 Level 3 ...

$$ \begin{array}{ccc} {\text { Factor B }} \\ \hline \text { Level 1 } & \text { Level 2 } & \text { Level 3 } \\ \hline 104 & 111 & 95 \\ \hline 82 & 104 & 84 \\ \hline 81 & 74 & 104 \\ \hline 111 & 109 & 106 \\ \hline 112 & 115 & 110 \\ \hline 82 & 97 & 99 \\ \hline 112 & 108 & 92 \\ \hline 108 & 99 & 129 \\ \hline 117 & 104 & 120 \end{array} $$ (a) Determine whether there is significant interaction between factor $\mathrm{A}$ and factor $\mathrm{B}$ (b) If there is no significant interaction, determine whether there is a significant difference in the means for the three levels of factor A. If there is no significant interaction, determine whether there is a significant difference in the means for the three levels of factor $B$ (c) Draw an interaction plot of the data to support the results of parts (a) and (b). (d) If there is a significant difference in the means for the three levels of factor A, use Tukey's test to determine which pairwise means differ using a familywise error rate of $\alpha=0.05 .$ If there is a significant difference in the means for the three levels of factor B, use Tukey's test to determine which pairwise means differ using a familywise error rate of $\alpha=0.05 .$

$$
\begin{array}{ccc}
{\text { Factor B }} \\
\hline \text { Level 1 } & \text { Level 2 } & \text { Level 3 } \\
\hline 104 & 111 & 95 \\
\hline 82 & 104 & 84 \\
\hline 81 & 74 & 104 \\
\hline 111 & 109 & 106 \\
\hline 112 & 115 & 110 \\
\hline 82 & 97 & 99 \\
\hline 112 & 108 & 92 \\
\hline 108 & 99 & 129 \\
\hline 117 & 104 & 120
\end{array}
$$
(a) Determine whether there is significant interaction between factor $\mathrm{A}$ and factor $\mathrm{B}$
(b) If there is no significant interaction, determine whether there is a significant difference in the means for the three levels of factor A. If there is no significant interaction, determine whether there is a significant difference in the means for the three levels of factor $B$
(c) Draw an interaction plot of the data to support the results of parts (a) and (b).
(d) If there is a significant difference in the means for the three levels of factor A, use Tukey's test to determine which pairwise means differ using a familywise error rate of $\alpha=0.05 .$ If there is a significant difference in the means for the three levels of factor B, use Tukey's test to determine which pairwise means differ using a familywise error rate of $\alpha=0.05 .$

Key Concepts

Interaction Plot

An interaction plot is a graphical tool used to visualize the potential interaction between two factors in an ANOVA analysis. It typically displays the means of the response variable for one factor across the levels of the other factor. This visualization helps in identifying whether lines representing different groups are parallel (indicating no interaction) or non-parallel (suggesting an interaction effect).

Tukey's HSD Test

Tukey's Honest Significant Difference (HSD) test is a post hoc analysis used after an ANOVA to determine which specific group means differ from each other. It controls the familywise error rate and is particularly useful when performing multiple pairwise comparisons, ensuring that the overall probability of making a type I error remains within a specified alpha level, such as 0.05.

Interaction Effect

An interaction effect in the context of two-factor ANOVA refers to a situation where the effect of one factor on the dependent variable differs depending on the level of the other factor. Assessing interaction is crucial because a significant interaction can indicate that the factors do not influence the response variable independently, which might require a different interpretation or further analysis of the data.

Two-Factor ANOVA

Two-factor ANOVA is a statistical method used to determine the effect of two independent categorical variables (factors) on a dependent variable simultaneously. This technique not only assesses the main effects of each factor but also examines if there is an interaction effect between the factors, providing a comprehensive analysis of the sources of variation in the data.

Main Effects

Main effects refer to the independent impact of each factor on the dependent variable, disregarding the influence of the other factor. In a two-factor ANOVA, if the interaction effect is not significant, researchers can focus on the main effects to determine whether the differences among levels of one factor are statistically significant, thereby understanding which factor has a measurable impact on the response.

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