Question

The relationship between the t-stat and F-stat is actually the consequence of a relationship between the distributions Tn and F1,n. By playing around with code in R, find a relationship between these distributions. Include the code that led to your conjecture relating these two distributions. Again, this is a problem where a Google search immediately gives the answer. Resist this temptation; learning is about the journey, not just the destination.

Answer 1

t is defined as standard normal variate divided by sq. root of (Chi-Sq/ df).

The above equation can be re-written as -

as Chi-sq with 1 df = square of SNV.

This can be easily verified in R using the following -

set.seed(1234)

Norm1 <- rnorm(1)
Norm1_sq <- Norm1**2
Chi1_sq <- Norm1_sq

Norm2 <- rnorm(c(1:10)) #df = 10
Norm2_sq <- Norm2**2
Chi2_sq <- sum(Norm2_sq)

t_stat <- Norm1/((Chi_sq/10)**(0.5))

t_stat_sq <- t_stat ** 2

F_stat <- (Chi1_sq/1)/(Chi2_sq/10)

isTRUE(round(t_stat_sq,8) == round(F_stat,8))