Maity and Sherman (2006) considered the situation in two-sample testing for the
di erence in means when one variance is assumed to be known while the other
variance is treated as unknown. This problem arises in many real life situations,
for example, when one is interested in comparing a standard treatment with a
new treatment in medical studies. The variance for the standard treatment is
assumed to be known from historical data, and the variance for the new treatment
is unknown. Following the argument in Satterthwaite (1941, 1946), Maity
and Sherman (2006) obtained the con dence interval for the di erence in means
based on an approximate t-distribution. In this paper, a likelihood-based third
order asymptotic method is introduced to obtain the con dence intervals for the
di erence in means. Simulations are used to show that the proposed method has
better coverage property than Maity and Sherman’s t-method, especially when
the sample sizes are small.