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My hypothesis is: As the experimental count variable increases, the probability that the binary dependent variable equals 1 increases. I expect both the independent and dependent variables to be not normally distributed with extra values at 0 for the count and "no" for the binary variable.
What model should I plan to use?
There is no need for the independent variable to be normally distributed. However, if your dependent variable is binary, the first model people usually try is logistic regression. Logistic regression will give you this model with the $\beta_0$ and $\beta_1$ calculated for you.
$$ \log\frac{p}{1-p} = \beta_0 + \beta_1 * count $$
It may be necessary to check your residuals to determine if this model fits the data properly.
What do the residuals in a logistic regression mean?
Given your hypothesis, you are looking for descriptive statistics and not prediction. Your software should give you a p-value for each $\beta$. If the p-value for $\beta_1$ is significant ($p <= 0.05$), then you have evidence of a relationship.
The last step is to state exactly what the relationship is. The easiest interpretation is this version of the model
$$ \frac{p}{1-p} = e^{\beta_0 + \beta_1 * count} = e^{\beta_0} e^{\beta_1 * count} $$
The term on the left is called the "odds ratio". Then you can say with count=0, the odds of success is $e^\beta_0$. As the count increases, $\beta_1$ is a multiplicative effect on the odds ratio and you would say: The odds of success increases $e^{\beta_1}$ with each additional count. For example, if $e^{\beta_1}=1.15$ then the odds of success increases 15% with each additional count.
There are many resources available describing the technique. Here are a few giving more details.
http://www.appstate.edu/~whiteheadjc/service/logit/intro.htm http://www.ats.ucla.edu/stat/r/dae/logit.htm http://www.unm.edu/~schrader/biostat/bio2/Spr06/lec9.pdf http://www.unm.edu/~schrader/biostat/bio2/Spr06/lec11.pdf
