## Interpreting confidence bands on interaction terms

Dear all,
I have a model with three variables a dependent (dep), a binary independent variable (bin) and a contin. independent variable (con). I am basically interested in whether the effect of con on dep is heterogeneous with respect to bin.

I regress:
dep=alpha0+alpha1*con+alpha2*bin+alpha3*con X bin + u

alpha1 is positive and significant, alpha3 is negative and signifcant. So I conclude that the effect is heterogeneous.

I then look at size of the effect of con on dep for the two groups:
1. For the baseline group (bin=0) the coefficient is: alpha1 and the standard error is the standard error on alpha1.
2. For the alternative group (bin=1) the effect should be: alpha1+alpha3 and the standard error is their combined standard error se(alpha1+alpha3).

My problem is now: when i compute confidence bands for the effects for each groups, the bands clearly overlap. So normally I would say that they are not statistically different. But my interaction term is very significant. What is wrong in my interpretation?