Hypothesis Tests

Welch t-Test

Compares the means of two independent samples without assuming equal variances.

Statusvalidated

Version1.0.0

Minitab equivalentStat > Basic Statistics > 2-Sample t (Do not assume equal variances)

Description

The Welch t-test is a modification of the two-sample t-test that does not assume equal variances. It uses the Welch-Satterthwaite approximation for degrees of freedom, which are generally non-integer. The Welch t-test is the preferred choice when an F-test or Levene test indicates unequal variances.

Formulas

t = \frac{x ˉ _{1} - x ˉ _{2}}{\frac{s _{1}^{2}}{n _{1}} + \frac{s _{2}^{2}}{n _{2}}}

Test statistic with separate variance estimates

df = \frac{( \frac{s _{1}^{2}}{n _{1}} + \frac{s _{2}^{2}}{n _{2}} ) ^{2}}{\frac{( s _{1}^{2} / n _{1} ) ^{2}}{n _{1} - 1} + \frac{( s _{2}^{2} / n _{2} ) ^{2}}{n _{2} - 1}}

Welch-Satterthwaite degrees of freedom (generally non-integer)

H₀

H_{0} : μ_{1} = μ_{2}

Null hypothesis: The population means are equal

Assumptions

Both samples are normally distributed
Samples are independent

Limitations

Slightly less powerful than the pooled t-test when variances are truly equal
Satterthwaite degrees of freedom are an approximation

References

Welch, B. L. (1947), The generalization of 'Student's' problem when several different population variances are involved, Biometrika, 34(1–2), 28–35
Satterthwaite, F. E. (1946), An approximate distribution of estimates of variance components, Biometrics Bulletin, 2(6), 110–114