Separate independent and paired designs

For independent samples with known variances, z=[(mean1-mean2)-Δ0]/√(σ1²/n1+σ2²/n2). For normal populations with unknown but equal variances, pool: sp²=[(n1-1)s1²+(n2-1)s2²]/(n1+n2-2). Then t=[(mean1-mean2)-Δ0]/[sp√(1/n1+1/n2)] with df=n1+n2-2. Equal variance is an assumption, not a consequence of independent sampling.