Revisiting the difference between two group means

In an earlier section, we described confidence intervals and tests about the difference between two group means, µ- 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 µ- 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.