3. Suppose Y n1 x nk b k1 n1 ;...

3. Suppose Yð Þ n1 ¼ xð Þ nk bð Þ k1 þ «ð Þ n1 ; where e ~

N(0,s2 I) and x has full column rank.

(a) Show that T ¼

ℓ

0 b^  b

S2 ℓ

0 x0 ð Þ x 1

ℓ

1=2 ;

for any conformable ℓ 6¼ 0 , has a t-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(0,1) random variable in the numerator, and the square root of a w2 random variable divided by its degrees of freedom in the denominator.)

(b) Using the fact that T has a t-distribution with (n  k)

degrees of freedom, define a random interval (Z1, Z2)

which satisfies Pðℓ

0 b 2 ðz1; z2ÞÞ ¼ :95 when n  k

¼ 25. (Hint: Define your z1 and z2 variables as appropriate functions of b^; S2

, which will be suggested by a transformation of P(t ∈ (tℓ, th)) ¼ .95), for outcomes t of T.