In an experiment, you measure the popularity of two brands of soft drinks (factor A), and for each brand you test males and females (factor B). The following table shows the main effect and cell means from the study:

Factor A
Level A1: Level A2:
Brand X Brand Y

Level B1: 14 23
Males
Factor B_______________________________
Level B2: 25 12
Females

(a) Describe the graph of the interaction when factor A is on the X axis.(b) Does there appear to be an interaction effect? Why? (c) What are the main effect means and thus the main effect of changing brands? (d) What are the main effect means and thus the main effect of changing gender? (e) Why will a significant interaction prohibit you from making conclusions based on the main effects?

(a) To describe the graph of the interaction when factor A is on the X axis, we need to plot the cell means for each combination of factor A and factor B.

Here is the graphical representation:

| Level A1 Level A2
-----------------------------------------------
Level B1 | 14 23
Level B2 | 25 12

In this graph, the x-axis represents the levels of factor A (Brand X and Brand Y), and the y-axis represents the cell means. We can plot two bars, one for each level of factor A, and each bar will have two sections representing the levels of factor B (males and females).

(b) Based on the graph, we can observe that the heights of the bars for each level of factor A are not consistent. The difference in heights between the two bars indicates that there is an interaction effect.

(c) The main effect means represent the average popularity across all levels of the other factor. In this case, the main effect means for factor A can be calculated by averaging the cell means across each level of factor A.

Main effect means for factor A:
- Level A1 (Brand X): (14 + 25) / 2 = 19.5
- Level A2 (Brand Y): (23 + 12) / 2 = 17.5

The main effect of changing brands is that Brand X (Level A1) has a higher average popularity (19.5) compared to Brand Y (Level A2) with a lower average popularity (17.5).

(d) Similarly, the main effect means for factor B can be calculated by averaging the cell means across each level of factor B.

Main effect means for factor B:
- Level B1 (Males): (14 + 23) / 2 = 18.5
- Level B2 (Females): (25 + 12) / 2 = 18.5

The main effect of changing gender is that there is no difference in the average popularity between males (Level B1) and females (Level B2), as both have an average popularity of 18.5.

(e) A significant interaction indicates that the effect of one factor (e.g., brand) on the outcome (popularity) depends on the level of another factor (e.g., gender). In other words, the effect of one factor is not consistent across different levels of the other factor.

If there is a significant interaction, we cannot solely rely on the main effects to draw conclusions because the interaction effect indicates that the relationship between the factors is more complex. The main effects might not capture the full picture of how each factor influences the outcome. Thus, considering the interaction effect is crucial for a comprehensive understanding of the relationship between the factors.