Correlation coefficient contradicts t-test I am looking at the probability of opting into a program based on a continuous variables $X_1$, $X_2$, $X_3$, etc.  When I divide the sample into people who opted in and did not opt in and do a t-test I find that the mean of all the $X$'s for those opting in is significantly different from the mean for those not opting in.
However, when I look at a correlation matrix for the $X$'s and opting in I find really low correlation coefficients -- from between $.04$ and $.13$.  But my t-tests are significant at the 95%+ confidence level.
How do I square these two results?  
 A: First, there's no real contradiction. The statistical significance of any statistic is only partly due to its size, it's also a function of sample size. How many people did you have?
Second, since "opting in" is a binary variable, and since you are treating it as a dependent variable with multiple independent variables, you really want logistic regression, not correlations.
A: I agree with Peter that there is no contradiction the t test and the correlations are telling you very different things.  First of all as Peter mentioned statistical significance depends on both the magnitude of the differnce and the sample size.  With a very large sample size small difference can be significant (even highly statistically significant).  
Now correlation between the variables measures whether or not the move together in a linear fashion.  It may be that if you have paired data for X$_1$ and X$_2$ that they don't tend to increase of decrease together or in the sense of negative correlation have X$_1$ decreasing while X$_2$ increases.  So for your data the magnitude of this tendency is low or non existent.
Now two variable can have very different means and zero correlation such as if
X$_1$(k)= A + ε(k)  and X$_2$(k)= B + η(k) where the ε(k) and η(k) are uncorrelated 0 mean noise terms that are also uncorrelated with each other.  
Suppose A-B>0.  Then for sufficiently large n (how large n has to be depends on the magnitude of the difference between A and B) the t test will say that the mean of X$_1$is statistically significantly different from the mean o9f X$_2$. But X$_1$ and X$_2$ are uncorrelated. This is like your situation.
On the other hand suppose X$_1$(k) = X$_2$(k) + ε(k)  where the ε$_s$ are zero mean independent gaussian noise terms their means will be the same but they could be highly correlated with the degree of correlation dependent on the variance of ε(k).
A: I don't agree that you necessarily will gain much from logistic regression, given my understanding of your research questions (To what degree do OptIns differ from NonOptIns?  Do we see more than chance differences?).  You've already determined using your own criteria that, for each of 3 X's, there is a statistically significant group difference but a weak one.  The weakness of the difference can be expressed in the mean difference, the standardized mean difference, or the point-biserial correlation with the OptIn variable, which is the technical name for the type of correlation you've calculated.  
I question your need for logistic regression because you haven't said anything about wanting to see to what degree the 3 X's can jointly predict the outcome; about learning the relative importance of each when the other two are controlled; or about assigning predicted probabilities of opting in for each person.  Those (among others) are the sorts of things logistic regression would give you.
