In hypothesis testing, which test statistic uses separate numerator and…

Concept

A sampling distribution may be indexed by one or more degrees-of-freedom parameters. These parameters describe how much independent information remains after estimating constraints.

When a statistic is formed as a ratio of two independently scaled variance estimates, its reference distribution tracks separate degrees of freedom for the numerator and denominator.

Application

An F statistic compares two mean-square or variance estimates as a numerator-to-denominator ratio.

Its reference distribution is written as F(ν1, ν2), where ν1 is the numerator degrees of freedom and ν2 is the denominator degrees of freedom.

Contrast

• The t distribution is indexed by a single degrees-of-freedom parameter.

• The standard normal Z distribution has no degrees-of-freedom parameter.

• The chi-square distribution is indexed by a single degrees-of-freedom parameter.

Cross-check

In one-way ANOVA, the statistic is a ratio of the between-group mean square to the within-group mean square; these terms carry degrees of freedom k − 1 and N − k, respectively.

Result

Therefore, the required test statistic is F.