A squared t variable

Topic: Statistical inference · more questions on this topic
‹ The expectation of the sample variance

Using only the definitions of the t and F distributions, show that if T\sim t_m then T^2\sim F_{1,m}. Hence explain why a two-sided t test at level \alpha and an F test with one numerator degree of freedom reject on the same samples.

Worked answer (click to show)

Write T=Z/\sqrt{V/m} with Z standard normal, V\sim\chi^2_m and the two independent. Then

T^2=\frac{Z^2/1}{V/m}.

Z^2 is \chi^2_1 by definition, and it is independent of V, so T^2 is F_{1,m}.

The two-sided t test rejects when |T|\ge t_{m;1-\alpha/2}, that is when T^2\ge t_{m;1-\alpha/2}^2. Because \mathbb{P}(T^2\le c^2)=\mathbb{P}(-c\le T\le c), the quantile F_{1,m;1-\alpha} equals t_{m;1-\alpha/2}^2. For example, t_{4;0.975}^2=2.776^2=7.71=F_{1,4;0.95}. The two tests reject on the same samples.