Pr(Z>|z|) values and the level of significance If your logistic regression fit has coefficients with the following attributes, do you look at the values of Pr(Z>|z|) are smaller than 0.95 to determine whether that variable is needed at a 5% level of significance? 
ie. If Pr(>|z|) is 0.964, this variable is not needed at 5% significance.

 A: Firstly, the p-value given for the Z-statistic would have to be interpreted as how likely it is that a result as extreme or more extreme than that observed would have occured under the null hypothesis. I.e. 0.96 would in principle mean that the data are providing very little evidence that the variable is needed (while small values such as, say, $p\leq 0.05$ would provide evidence for the likely relevance of the variable, as pointed out by others already). However, a lack of clear evidence that the variable is needed in the model to explain the this particular data set would not imply evidence that the variable is not needed. That would require a difference approach and with a very larege standard error one would not normally be able to say that the variable does not have an effect. Also, it is a very bad idea to decide which variables are to be included in a model based on p-values and then fitting the model with or without them as if no model selection had occurred.
Secondly, as also pointed out by others, when you get this huge a coefficient (corresponds to an odds ratio of $e^{-14.29}$) and standard error from logistic regression, you typically have some problem. E.g. the algorithm did not converge or there is complete separation in the data. If your model really did only include an intercept, then perhaps there are no events at all, and all records did not have an outcome? If so, then a standard logistic regression may not be able to tell you a lot. There are some alternatives for such sparse data situations (e.g. a Bayesian analysis including the available prior information).
A: You are using the normal approximation and specifically the Wald test so you do what you would do in a regular t-test. That is, you reject the null hypothesis if the probability of the event $\left\{Z \geq |z| \right\}$ is lower than the conventional threshold of $0.05$. Alternatively you fail to reject the null hypothesis if your p-value is not small enough.
A: The value of the coefficient and its large standard error suggest that what we are seeing here is separation or the Hauck-Donner effect which has its own tag hauck-donner-effect which has a clear and helpful wiki excerpt. I think therefore the debate about $t$ versus $z$ is a red herring. Profile likelihood would be the way to go or reformulating the problem.
