In this paper new binary sequence families $\mathcal{F}^k$ of period $2^n-1$ are constructed for even $n$ and any $k$ with ${\rm gcd}(k,n)=2$ if $n/2$ is odd or ${\rm gcd}(k,n)=1$ if $n/2$ is even. The distribution of their correlation values is completely determined. These families have maximum correlation $2^{n/2+1}+1$ and family size $2^{3n/2}+2^{n/2}$ for odd $n/2$ or $2^{3n/2}+2^{n/2}-1$ for even $n/2$. The construction of the large set of Kasami sequences which is exactly the $\mathcal{F}^{k}$ with $k=n/2+1$ is generalized.