Ultra-relativistic heavy ion collisions provide a unique opportunity for study of a new state of matter known as the Quark Gluon Plasma (QGP) under laboratory conditions. The Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory is an accelerator designed to create and study such plasma. The azimuthal anisotropy of particle production in these collisions is a unique tool for studying the properties of the QGP. It is quantied by the second harmonic v<sub>2</sub> of the Fourier expansion of the azimuthal distribution of emitted particles. This elliptic flow is created by pressure gradients caused by the geometrical anisotropy of the initial collision zone, and transformed into momentum anisotropy of produced particles during the hydrodynamic expansion of the QGP medium. In this study, precision measurements of v<sub>2</sub> are presented for data collected by the PHENIX detector for Au+Au collisions at sqrt(sNN) = 200 GeV in Run 7. The measurements are performed for charged hadrons with transverse momentum of 0.4. 1.95 GeV/c, for the collision centrality range 0-50%. To study the influence of non-flow effects, measurements were also obtained with event planes separated from the PHENIX central arms by different pseudorapidity gaps. The v2 measurements indicate large values of v2 which are compatible with QGP formation in Au+Au collisions at sqrt(sNN) = 200 GeV. Measurements for various event planes show results which are consistent with each other, suggesting a negligible role for non-flow effects in the momentum and centrality range used in this analysis. Detailed comparisons of the v2 measurements obtained with different event planes, as well as comparisons to the results obtained by the STAR collaboration, indicate overall good agreement; relatively small differences in the most central collisions can be understood in terms of a systematic uncertainty in the centrality estimates by the PHENIX and STAR experiments.The precision measurements presented here provide an important experimental basis for reliable extraction of QGP transport coefficients such as viscosity from v<sub>2</sub> measurements.