The effect of the quality of starting powders on the microstructure and superconducting properties of in-situ processed Fe-sheathed MgB2 tapes has been investigated. Three different types of commercial atomized spherical magnesium powder and two different purities of amorphous boron powder were employed. When using the 10-micrometre magnesium as precursor powders, the Mg reacted with boron more uniformly and quickly, thus the uniformity of the fabricated MgB2 was improved and the grain size of the MgB2 was decreased, hence significant critical current density (Jc) enhancements were achieved for MgB2 tapes. Jc at 4.2 K for MgB2 tapes made from the 10 um Mg and high purity boron powders was at least a factor of ten higher than values measured for MgB2 samples made from all other starting powders. At 20 K, 5 T, the typical Jc values of the tapes were over 1.0x10^4 A/cm^2 and were much better than those of tape samples reported recently.