Contact processes -- and, more generally, interacting particle processes -- can serve as models for a large variety of statistical problems, especially if we allow some simple modifications that do not essentially complicate the mathematical treatment of these processes. In a forthcoming paper we shall begin a statistical study of the supercritical contact process $xi_t ^{\{0\}$ that starts with a single infected site at the origin and is conditioned on survival. There we shall consider the simplest statistical problem imaginable, that is, to find an estimator of the parameter of the process based on observing the set of infected sites at a single time $t$. We shall show that this estimator is consistent as $t \rightarrow \infty$ and establish its limit distribution after proper normalization. First, however, we must push some known properties of the contact process a little further. The present paper is devoted to these matters. In particular, we study the convex hull of the set of infected sites for the conditional $xi_t ^{\{0\}$ process as well as its spatial correlation. We find that under some restrictions this correlation decays faster than any negative power of the distance.