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Inferences about a mean vector Readings from Johnson 10.2 The Central Limit Theorem: Let be independent observations for a population with mean ) ,....... , ( 2 1 N x x x µ and variance covariance Σ Then: ) ˆ ( N is approximately ) , 0 ( Σ p N and () ( Σ ˆ ˆ 1 ' N ) is approximately 2 p χ For p N >>>>>>>>>> Let’s review the univariate one sample test. 0 0 0 : : = a H H The appropriate statistic is n t σ ˆ ˆ 0 = Reject if For a 100*(1- α ) confidence interval: n t n t n n ˆ * ˆ ˆ * ˆ 1 , 2 / 0 1 , 2 / + Inferences about a mean vector Lecture.doc 1

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Rejecting Ho when |t| is large is equivalent to rejecting Ho when t 2 is large ) ˆ ( ) ˆ )( ˆ ( ˆ ) ˆ ( 0 1 2 0 2 2 0 2 µ σ = = n n t Reject if 2 1 , 2 / 0 1 2 0 ) ˆ ( ) ˆ )( ˆ ( > n t n α A generalization of the squared distance is the multivariate analog. ) ˆ ( ˆ ) ˆ ( 0 1 ' 0 2 Σ = n T The statistic 2 T is called Hotelling’s 2 T in honor of Harold Hotelling who first obtained its sampling distribution. 2 T is distributed as p n p F p n p n , , ) ( ) 1 ( Inferences about a mean vector Lecture.doc 2
In summary: Let be a random sample from an N X X X ,....... , , 2 1 ) , ( Σ µ p N . Then with = = n i i x n 1 1 ˆ and () > Σ = > = = Σ = p n p p n p n i F p n p n n P F p n p n T P n , , 0 1 ' 0 , , 2 1 ' 1 ) ( ) 1 ( ) ˆ ( ˆ ) ˆ ( ) ( ) 1 ( ˆ ˆ 1 1 ˆ α One sample multivariate test of hypothesis. 0 0 0 : : = a H H At the level of significance reject Ho if: p n p F p n p n n T > Σ = , , 0 1 ' 0 2 ) ( ) 1 ( ) ˆ ( ˆ ) ˆ ( Example: Test the hypothesis [] 10 50 4 : 10 50 4 : ' ' 0 = a H H = 965 . 9 400 . 45 640 . 4 ˆ = Σ 628 . 3 627 . 5 810 . 1 627 . 5 798 . 199 002 . 10 810 . 1 002 . 10 879 . 2 ˆ = Σ 402 . 002 . 258 . 002 . 006 . 022 . 258 . 022 . 586 . ˆ 1 Inferences about a mean vector Lecture.doc 3

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Confidence regions and simultaneous comparisons of component means. A )% 1 ( 100 α confidence region for the mean of a p-dimensional normal distribution is the set determined by all µ such that p n p F p n p n n Σ , , 1 ' ) ( ) 1 ( ) ˆ ( ˆ ) ˆ ( Calculation of the axes of the confidence ellipsoid and their relative lengths. p n p F p n p n c n = Σ , , 2 1 ' ) ( ) 1 ( ) ˆ ( ˆ ) ˆ ( The direction and length of the ellipsoid are determined by: p n p i F p n p n n ± , , ) ( ) 1 ( λ Inferences about a mean vector Lecture.doc 4
Example: For n=42 , radiation from microwave ovens.

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