Differences between exponential distribution with very different n sizes

I need to test for the differences between three groups of observations (grouped along $$x$$-axis), which appear to follow an exponential distribution along the $$y$$-axis dimension (see example fig.). Tentatively, I used the MLE for $$\mu$$ and their respective 95% CI for each group and deemed them significantly different if their CI are non-overlapping. However, I'm not certain this is formally appropriate, particularly given that the sample sizes are markedly different.
$x$ axis = {groups a, b, c} shown with jitter); the sample size is $$n$$ in each case. The red line and pink shaded area depict the MLE $$\mu$$ and C.I., respectively, across all groups. Only the C.I. intervals between groups a & c are not overlapping (the triangle symbol is just a flag for the non-overlapping CIs, and the 'a' on upper-left is the subplot id)">

• Welcome to our site. Please explain what you mean by the "n-size" of a distribution. Indeed, since nothing in the graphic looks remotely like it is described by an exponential distribution, would you mind explaining what you mean by this term? – whuber Oct 30 '18 at 15:44
• n meaning sample size – Juan Vainas Oct 30 '18 at 16:03
• Okay. Are you trying to ask about the meaningfulness of comparing group means by evaluating whether confidence intervals are overlapping? We have some answers about that, including stats.stackexchange.com/questions/18215 and stats.stackexchange.com/questions/31657. – whuber Oct 30 '18 at 17:04
• Yes, exactly that – Juan Vainas Oct 30 '18 at 17:09