I am not sure if I've made a computational error, am failing to think something through (too late in the day), or just really don't understand confidence intervals as much as I was hoping I did, but I ran a linear model with Condition as a predictor (2 groups) and the slope representing the group difference was significantly different from zero:
b = 0.10599, t = 2.043, p = 0.045, 95% CIs [0.0023, 0.21]
However, when I compute the CIs for the two group means they overlap:
Experimental: M = .32, 95% CIs [.24, .39] Control: M = .21, 95% CIs [.14, .28].
What's going on? The overlapping CIs suggest to me that the means aren't significantly different, but I thought testing whether the mean difference differs significantly from zero is the same as testing whether the means differ significantly from each other?