If σ1 and σ2 were known...

Prob (  is within   ±  1.96     of   μ2 - μ1)   =   0.95

so a 95% confidence interval for µ2 - µ1 would be

  ±   1.96  

When σ1 and σ2 are unknown...

We must replace σ1 and σ2 by s1 and s2 in the confidence interval, and the constant '1.96' must be replaced by a slightly larger value from t-tables,

where the degrees of freedom for the t-value are

ν   =   min (n1−1,  n2−1)

(A more complex formula is available that gives a higher value for ν. It is slightly better but the difference is usually negligible.)

Example