Revisiting a confidence interval for two group means
The 95% confidence interval for the difference between two group means can be improved if the two standard deviations can be assumed to be equal.
Recommendation
It is usually best to avoid assuming equal standard deviations and use the earlier confidence interval.
Examples
Hypothesis tests
If it can be assumed that σ1 = σ2 = σ, the test for equal means can also be modified.
H0 : µ2 − µ1 = 0
HA: µ2 − µ1 ≠ 0
The test would be based on the test statistic
The p-value for the test is found from the tail area of the t distribution with (n1 + n2 - 2) degrees of freedom. However it is usually better to stick with the earlier test to avoid making assumptions about the variances.