Revisiting the difference between two group means
In an earlier section, we described confidence intervals and tests about the difference between two group means, µ2 - µ1. They can be improved if we can assume that
σ1 = σ2 = σ
Inference is still based on ,
but the equation for its standard deviation can be simplified
Confidence interval
A 95% confidence interval for µ2 - µ1 has the same general form as before,
but the standard deviation and the degrees of freedom for the t-value, ν, are different.
![]() |
degrees of freedom | |
---|---|---|
Allowing σ1 ≠ σ2 | ![]() |
min( n1 - 1, n2 - 1) |
Assuming σ1 = σ2 = σ | ![]() |
n1 + n2 - 2 |
If it can be assumed that σ1 = σ2, the confidence interval is usually narrower.
Example
The diagram below shows 95% confidence intervals obtained by the two methods.
The p-value for this test is found from the tail area of the t distribution with (n1 + n2 - 2) degrees of freedom.