Final answer:
To verify the relationship between a t-distributed variable T with ν degrees of freedom and an F-distributed variable X = T², we reference properties showing that T² follows an F-distribution with 1 and ν degrees of freedom. This relationship is due to the F-distribution values being squares of t-distribution values.
Step-by-step explanation:
To verify that if T has a t-distribution with ν degrees of freedom, then X = T² has an F-distribution with ν1 = 1 and ν2 = ν degrees of freedom, we refer to the properties of the distributions. By definition, a variable T that is t-distributed with ν degrees of freedom, when squared, results in a variable that follows an F-distribution with (1, ν) degrees of freedom. This relationship relies on the fact that the t-distribution is related to the F-distribution such that the square of a t-distributed variable (with ν degrees of freedom) is F-distributed with df(num) = 1 and df(denom) = ν.
The F-distribution has properties where its values are the squares of corresponding values of the t-distribution, and thus it aligns with the assertion provided. As the degrees of freedom for the numerator (which is 1 in this case) and for the denominator (which equals the degrees of freedom of the t-distribution) get larger, the F-distribution curve approximates the normal distribution.