# notes10 - Stat 430/Math468 Notes#10 Inference about a Mean...

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Stat 430/Math468 – Notes #10 Chapter 5 Inference about a Mean Vector Suppose are random sample from a normal population . The sample mean and sample variance-covariance matrix are, respectively, 1 ,..., n XX (, ) p N μΣ 1 12 ( ... ) n n =+ + + X X and 1 1 1 () n jj n j = =− SX X X ' X . Hypotheses: 00 : H = μμ vs , where 0 : a H 10 ( ,..., )' p μ = μ Test statistic: Hotelling’s 21 ' ( Tn X μ μ ) Null distribution: Under , 0 H 2 , (1 ) ~ pn p np T F . [Equivalently, 2 , ~ ) TF ] Decision Rule (Critical value method. For P-value method, see remark 1.): Suppose the observed sample mean vector is x and sample variance-covariance matrix is . Then at significance level S α , reject if 0 H , ) ' ( F ) −> x μ Sx μ where , F is the upper 100 th percentile of the , F distribution. Remark : 1. Suppose the observed is . Then the P-value = 2 T 2 obs t 2 , ) obs PF t ⎛⎞ > ⎜⎟ ⎝⎠ . 2. is invariant under linear transformation of the data (e.g. changing the units of measurements: 2 T o F Æ o C).

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notes10 - Stat 430/Math468 Notes#10 Inference about a Mean...

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