Two-Way ANOVA with non-sig interaction effect: is it reasonable to argue for an interaction from simple effects? This is in the context of marking an undergraduate sociology essay in which the student has not found the predicted significant interaction effect in a Two-Way ANOVA.
The diagram below is schematic and doesn't depict the actual data. Imagine that the interaction effect is non-significant. The student has said that because the effect of task is significant in the treatment group but non-significant in the control group, there's some evidence that there really is an interaction effect. 

I sense there is something fishy about this but I am struggling to come up with a precise explanation of why.
 A: The student wrote

because the effect of task is significant in the treatment group but
  non-significant in the control group, there's some evidence that there
  really is an interaction effect.

By itself, this is not very good evidence of an interaction effect, for reasons others have identified:


*

*p in treatment group could be 0.049 and in control group 0.051, this would not be evidence of anything in particular.

*Similarly, the effect size could be identical and, thus, the interaction exactly 0, and yet the p could be significant in one and not in the other due to different group sizes.
What, then, would be evidence of an interaction? That depends on whether you mean an important interaction or a significant one. The two are not the same. The importance of the interaction can be seen in a plot like the one in the question (only with real data!) or by examining the effect size in each group separately, or by looking at a regression with an interaction term and seeing how big that interaction term is (not how small its p-value is). The significance of the interaction can be examined by the p-value of the interaction term in the equation. 
In addition to the usual reasons for not using p-values to indicate things they don't indicate, I will note that interaction terms are notorious in that, if the main effects that go into the interaction are measured with error (as nearly everything is, but usually more so in subjects like sociology than subjects like chemistry) then the error on the interaction term is even larger. 
