3. Suppose Ylnxll = xlnxkl{3lkxll + elnxll, where e N([Oj, a2 I) and x has full column rank.

a. Show that T = l'(i3 - (3)

(S2l'(x'X)-ll)l/2'

for any conformable l 1= [01, has at-distribution with (n - k) degrees of freedom. (Hint: Transform T so that it is expressed as a ratio of two independent random variables, with an N(O, 1) random variable in the numerator, and the square root ofaX2 random variable divided by its degrees of freedom in the denominator.)