How to interpret negative 95% confidence interval? I have performed an unpaired t-test on two groups of data:
Group 1 (control): mean = 0.087 (n = 15) 
         SD = 0.028
Group 2 (treatment): mean = 0.115 (n=12) 
         SD = 0.042 
The t-test revealed a non-significant difference between the groups with the 95% confidence interval being -0.056 to 0.00068 
What does the negative confidence interval mean? I understand that I can't reject the null hypothesis because zero is contained within the CI range. But why is the range more "weighted" towards a negative value rather than a positive?
 A: For a two-sample t-test (paired or unpaired), what you are looking at is the difference between the means of the two samples. The 95% confidence interval is providing a range that you are 95% confident the true difference in means falls in. Thus, the CI can include negative numbers, because the difference in means may be negative.
For a very basic example, let's say that your control group has a mean of $1$ and your treatment group has a mean of $2$. The difference between these will be $-1$. When you calculate the confidence interval for the true difference in means (not just the sample difference), it will be centered on $-1$. The confidence interval (whatever it is) will by definition fall more in the negative side than the positive side. However, if you reversed the calculation and did treatment-control instead, you would get a range falling more in the positive side. It would not affect your final conclusion.
[EDIT] The numbers in the question got updated, but I'll leave this comment here for future reference: In the numbers you give (as whuber points out in a comment), your confidence interval should center on $-0.033$ because that's the difference in your sample means. Because it doesn't, there's likely some error in your calculations somewhere.
A: The range is "weighted" because the estimated differences in the means is not exactly zero. The CI of the difference is the point estimate +/- 1.96 * SE and it will only be symmetric about zero when the point estimate is zero. 
A: There is no inconsistency in difference between means and center point of confidence intervals. Both are 0.028
In simple terms, a negative confidence interval in this setting means that although observation is that mean of group 2 is 0.028 higher than group 1, the 95% confidence interval suggest that actually group 1 may be higher than group 2. 
A: Or does the CI definition change based on what we are using it for, as it only estimates our confidence at say 95% that the true value falls between the upper and lower limits of CI intervals
For example; 
In trying to figure out the effect of Y~ ßX + intercept, the 95% CI for X effect will give an estimate of true ß value (± 2.SE of ß)
Whereas in non inferiority study, the 95% CI is an estimate whether drug X (new) is non inferior than drug Y (baseline) by assigning a predefined margin M
Another is CI of means difference above
As such the negative or non negative values discussion have many meanings to it depending on how one uses 95% CI for?
