In Problems 58-64 you will show that the univariate t-statistic has a t-distribution with n − 1 degrees of freedom when

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In Problems 58 64 You Will Show That The Univariate T Statistic Has A T Distribution With N 1 Degrees Of Freedom When 1 (394.07 KiB) Viewed 21 times

In Problems 58-64 you will show that the univariate t-statistic has a t-distribution with n − 1 degrees of freedom when the population is normal. For these problems, suppose that X₁,..., Xn are iid N (μ, o2). It is possible to construct a nonrandom nxn matrix A with columns A₁,..., An such that A₁ = (1/√n,...,1/√√n)' and A'A = AA' = In where In is the n x n identity matrix. We will not prove this fact, but will assume that A is such a matrix. Note that for any i, j in {1,...,n}, Aj Aj = 0 if i j, and A; Ai 1. Define the vector X let Z = A'X, so that Z is an n-dimensional column vector, Z = (Z₁,,Zn)'. We can write (X₁, Xn)' and n X = Σj=1 AjZj. 58. Show that Z₁ = √n X. = =