Chapter 13 - Section 13.4 - Assess Your Understanding - Vocabulary and Skill Building - Page 665: 9

(a) The factor A main effect is calculated by taking difference of average of the dependent variable when factor A is high and low. It is computed as follows: \[\begin{align} & =\frac{20+14+1+3}{4}-\frac{12+8+6+7}{4} \\ & =\frac{38}{4}-\frac{33}{4} \\ & =1.25 \\ \end{align}\] If factor A increases from low to high level, the dependent variable increases by 1.25 units. The factor B main effect is calculated by taking difference of average of the dependent variable when factor B is high and low. It is computed as follows: \[\begin{align} & =\frac{6+7+1+3}{4}-\frac{12+8+20+14}{4} \\ & =\frac{17}{4}-\frac{54}{4} \\ & =-9.25 \\ \end{align}\] If factor B increases from low to high level, the dependent variable decreases by 9.25 units. The effect of one factor on other can be checked by using the interaction effect. Considering lower level of factor B, it can be seen that effect of factor A from low to high is \[\begin{align} & \frac{\text{high level of factor A}}{2}-\frac{\text{low level of factor A}}{2} \\ & =\frac{20+14}{2}-\frac{12+8}{2} \\ & =17-10 \\ & =10 \\ \end{align}\] So, at lower level of factor B, dependent variable changes by 10 units when factor A changes from low to high. Considering higher level of factor B, it can be seen that effect of factor A from low to high is \[\begin{align} & \frac{\text{high level of factor A}}{2}-\frac{\text{low level of factor A}}{2} \\ & =\frac{1+3}{2}-\frac{6+7}{2} \\ & =2-6.5 \\ & =-4.5 \\ \end{align}\] So, at higher level of factor B, dependent variable decreases by 4.5 units when factor A changes from low to high. Considering lower level of factor A, it can be seen that effect of factor B from low to high is \[\begin{align} & \frac{\text{high level of factor B}}{2}-\frac{\text{low level of factor B}}{2} \\ & =\frac{6+7}{2}-\frac{12+8}{2} \\ & =6.5-10 \\ & =-3.5 \\ \end{align}\] So, at lower level of factor A, dependent variable decreases by 3.5 units when factor B changes from low to high. Considering higher level of factor A, it can be seen that effect of factor B from low to high is \[\begin{align} & \frac{\text{high level of factor B}}{2}-\frac{\text{low level of factor B}}{2} \\ & =\frac{1+3}{2}-\frac{20+14}{2} \\ & =2-17 \\ & =-15 \\ \end{align}\] So, at higher level of factor A, dependent variable decreases by 15 units when factor B changes from low to high. It is seen that changes in dependent variable for factor A depend on level of factor A. So, there is interaction effect among both factors. (b) The interaction plot is shown below:

The interaction plot also shows that there is interaction between factors.