(a) Ņull hypothesis, Ho : = 3, β2 = 3 Alternative hypothesis, Hi at least one Bi 3 Under Ho, (b1-3. b2-3) (XX) σ 2 ) ~ Since ơ2 is unknown, so ơ2 is estimated by ee where' _ ~ X39 and bı - 3 29 (b1-3, b2-3) (XX)ơ-2 (bz-3) and ee are independent so bı -3 ее bi - 3 b2-329 3(b- 3)2(b -3) (b2 -3) 2(b2 - 3)2 F2,29 under Ho
Here, b 2, b2 3, e'e-60 then the value of F 0.725
p- value-PF > 0.725 FF2,290.4929 >0.05 Hence we fail to reject Ho at 5% level of significance Var(bi - b2)1)Disp (1 ee ee H by where Var(bi b2)77(60/29) 14.4828, se (bi-b2) =、〈Var(b1-b2) = 3.80056 Now, under Ho. the test statistic. t = ere σ2 is unknown so we estimate σ ~ 29 ее 29 bi - b2 e(-b229 se(01 Value of test statistic. t = (2-3)/3·8056 =-0.2628 p-value = 2P(t > 0.2628|t ~ t29) = 0.7946 > 0.05 Hence we fail to reject Ho at 5% level of significance Hence we use same test statistic of (b) and get same conclusion as (b)
(d) 90% C.I.for β2-βι is (b2 - bi - to.o5,29 se (bi -b2), b2 - b to.05,29 se(bi - b2)) -5.4661, 7.4661); where, to.05,29 от 0.05,29 Se (01 1.6991