In this paper, for an odd prime p and i=0,1, we investigate the cross-correlation between two decimated sequences, s(2t+i) and s(dt), where s(t) is a p-ary m-sequence of period pⁿ-1. Here we consider two cases of ${d}$, ${d=\frac{(p^m +1)^2}{2} }$ with ${n=2m}$, ${p^m \equiv 1 \pmod{4}}$ and ${d=\frac{(p^m +1)^2}{p^e + 1}}$ with n=2m and odd m/e. The value distribution of the cross-correlation function for each case is completely determined. Also, by using these decimated sequences, two new p-ary sequence families of period ${\frac{p^n -1}{2}}$ with good correlation property are constructed.