Using the Drude-Boltzmann semiclassical transport theory, we calculate the weak-field Hall resistance of a two-dimensional system at low densities and temperatures, assuming carrier scattering by screened random charged impurity centers. The temperature dependent 2D Hall coefficient shows striking non-monotonicity in strongly screened systems, and in particular, we qualitatively explain the recent puzzling experimental observation of a decreasing Hall resistance with increasing temperature in a dilute 2D hole system. We predict that the impurity scattering limited Hall coefficient will eventually increase with temperature at higher temperatures. PACS Number : 71.30.+h; 73.40.Kp; 73.40.Qv The behavior and the properties of the apparent two dimensional (2D) "metallic" phase continue to attract substantial attention [1] from experimentalists and theorists alike, even a decade after its original discovery [2]. In particular, the original observations on the strong metallic (i.e. dρ/dT > 0) temperature dependence of the 2D resistivity, ρ(T ), where the resistivity may increase by as much as a factor of 3 − 4 for a modest increase in temperature (e.g. T = 100mK − 3K) were followed by intriguing observations of large magnetoresistance in an applied parallel magnetic field. Phenomenologically the observed "metallicity", defined as the maximum temperature induced enhancement of ρ(T ), exhibits strong system dependence, with 2D p-GaAs hole system being the most metallic and 2D n-GaAs electron system being the least metallic with the 2D Si-based electron systems having intermediate metallicity. This system-dependent variation can be understood on the basis of our theoretical prediction [3] that the metallicity arises from 2D screening properties, and is therefore controlled in the zeroth order theoretical prescription by the dimensionless parameters q T F /2k F and T /T F , where q T F , k F , T F are respectively 2D Thomas-Fermi screening wave vector, the Fermi wave vector, and the Fermi temperature [4]. Since q T F /2k F ∝ mn −1/2 and T /T F ∝ mn −1 in 2D, the metallicity increases with increasing (decreasing) carrier effective mass m (carrier density n). This is why metallicity is stronger at lower carrier densities and/or in higher effective mass semiconductor systems.The idea [5] of the strongly temperature dependent screened charged impurity effective disorder being the qualitative reason underlying the striking metallic behavior of dilute 2D carrier systems has come to be known as the "screening theory" since the screening-induced regularization of the bare Coulombic impurity disorder (in contrast to zero-range white noise disorder) intrinsic to semiconductor systems is the crucial physical mechanism in this theory. The screening theory has been applied, with reasonable qualitative success, to explain the temperature and density [3,5,6] dependence as well the parallel magnetic field dependence [7] of the 2D "metallic" resistivity [3,5]. Motivated by a puzzling recent experimental observation [8], we develop in th...