I am confused about an apparent contradiction between t-test and 1-way ANOVA in one particular case - please suggest a way to think about it.
Suppose I want to compare some parameter between 3 groups, but I am mostly interested in comparison between group 1 and 2. The collected data looks like the graph below (dots: individual data points, lines: means +/- 95% CI). T-test of group 1 versus group 2 is highly significant (p<0.01). But the 1-way ANOVA for all 3 groups is non-significant (p=0.10). Intuitively, one can quite clearly see that groups 1 and 2 are different. But the addition of the group 3 with high variability obscures this fact. I feel that statistics in this case obscures the common sense.
As an illustration consider this thought experiment. Imagine that I first collected only group 1 and 2, did the t-test, and concluded that these populations have different means. Then, I added group 3 (which is not even that important in the real experiment). Now, the formally correct test would be 1-way ANOVA, and the conclusion now is that "there is not enough evidence that populations have different means". But from the point of view of common sense, I don't understand how addition of the third group can change the previously established fact that populations 1 and 2 have different means.
Could you please suggest the way to reconcile the statistical and practical conclusions in this case?
Maybe there is some justification of using t-test instead of ANOVA in such cases?