What is the best way to test for a relationship between a binary variable and a continuous variable? Suppose I have a dataset with a continuous numerical variable $x$ and a binary numerical variable $y$ (with values 0 or 1).
How can I test the null hypothesis that the value of $x$ has no effect on the binary outcome $y$?
My idea was to use a two-sample t-test where one sample is the values of $x$ when $y=0$ and another sample is the values of $x$ when $y=1$. Would this be appropriate?
 A: Your approach would not be the best.  It treats the predictor as the outcome, which is dubious scientifically speaking.  After all, your question is about $E(y \vert x)$ not $E(x \vert y)$.
The best way would be to perform a logistic regression.  I'm not going to get into the details here, there are lots of resources for learning about logistic regression.  Here is a small example in R.
I've generated a continuous predictor and a binary outcome.  In the plot below, I've binned the predictor and computed the average of the outcomes.  As the predictor increases, we seem to get more outcomes where $y=1$.

We can perform a test of association by fitting a logistic regression.  In R,
model=glm(y~x, data=my_data, family=binomial())
summary(model)
> summary(model)

Call:
glm(formula = y ~ x, family = binomial(), data = my_data)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.0942  -0.8878  -0.8205   1.4394   1.8072  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) -0.79111    0.06868 -11.518  < 2e-16 ***
x            0.20272    0.06929   2.926  0.00344 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 1244.5  on 999  degrees of freedom
Residual deviance: 1235.8  on 998  degrees of freedom
AIC: 1239.8

Number of Fisher Scoring iterations: 4


Look at the row for x in the summary.  It has an associated p value and an estimate for the effect.  The effect is the log-odds ratio (again, something you can easily read up on).  If this estimate is positive, then as the predictor increases then so too will the probability we see a positive outcome.  If it is negative, the opposite is true.
