  We study the quadratic family of one-dimensional maps $f_a (x) = a - x^2$. We
conduct comprehensive numerical analysis of collections of finite orbits of the
critical point, computed for intervals of parameter values using rigorous
numerical methods. We use the computer to explicitly construct a collection of
several thousand parameter intervals, contained in $\Omega=[1.4, 2]$, that are
proved to have a specific so-called escape time, which roughly means that some
effectively computed iterate of the critical point taken over all the
parameters in that interval has considerable width in the phase space. In
particular, we compute a rigorous lower bound on this width, in addition to the
upper bound. We investigate the effect of certain constraints imposed on the
numerical computations upon the resulting collection of intervals.
Additionally, we illustrate and discuss the distribution of the computed
intervals in the parameter space. The purpose of our work is to establish
grounds for further numerical computation of a lower bound on the measure of
stochastic parameters in $\Omega$. The source code of the software and the data
discussed in the paper are freely available at
http://www.pawelpilarczyk.com/quadr/. This web page also allows carrying out
some limited computations. The ideas and procedures introduced in the paper can
be easily generalised to apply to other parametrised families of dynamical
systems.
