Answer Happy

In Problems 58-64 you will show that the univariate t-statistichas a t-distribution with n − 1 degrees of freedom when thepopulation is normal. For these problems, suppose that X1,··· ,Xnare iid N(μ, σ2). It is possible to construct a nonrandom n×nmatrix A with columns A1, · · · , An such that A1 = (1/√n,···,1/√n)′ and A′A = AA′ = In where In is the n × n identity matrix.We will not prove this fact, but will assume that A is such amatrix. Note that for any i, j in {1,··· ,n}, A′iAj = 0 if i ̸= j,and A′iAi = 1. Define the vector X = (X1,··· ,Xn)′ and let Z = A′X,so that Z is an n-dimensional column vector, Z = (Z1, · · · , Zn)′.We can write X = 􏰄nj=1 AjZj.
√
58.ShowthatZ1= nX.
59. Show that E[Zj] = 0 for j = 2,··· ,n.
60. Show that Cov(Z) = σ2In.
61. Use the results above to show that Z1, · · · , Zn areindependent, and normally distributed
with variance σ2, with E[Z1] = √nμ and E[Zj] = 0 for j = 2,···,n.
62. Show that X − X ̄ = 􏰄nj=2 AjZj.
63. Show that 􏰄ni=1(Xi − X ̄)2 = (X − X ̄)′(X − X ̄) = 􏰄nj=2Zj2.
64. Use the results above to show that t = X ̄−μ has the tdistribution