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 (367.22 KiB) Viewed 26 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 (u, 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 × 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}, A; A; = 0 if i ‡ j, and AA₁ 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 = Σj=1 AjZj. = =

62. Show that X - X = 63. Show that -2 AjZj. ₁ (Xi − X)² = (X - X)'(X - X) = Σ₁=2 2².