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STAT 501 Spring 2005 Assignment 2 NAME __________________ Reading Assignment: Chapter 5, and Sections 6.3 in Johnson & Wichern. Written...

"How do you do 5e and 5f of Iowa State's STAT501 hw2s05? What test do you run? "

STAT 501 Assignment 2 NAME __________________ Spring 2005 Reading Assignment : Chapter 5, and Sections 6.1-6.3 in Johnson & Wichern. Written Assignment : Due Monday, February 14, in class. You should be able to do the first four problems without a computer, but use computer packages in any way you desire to answer these questions. 1. The following data consist of measurement made on the levels of three liver enzymes (U/L): aspartate aminotransferase (X 1 ), alanine aminotransferase (X 2 ), and glutamate dehydrogenase (X 3 ) in n=10 patients diagnosed with aggressive chronic hepatitis. Patient X 1 X 2 X 3 1 31 63 4 2 32 56 6 3 50 59 9 4 56 72 7 5 39 87 9 6 46 95 8 7 29 57 5 8 40 50 3 9 29 44 4 10 24 42 3 These are part of a larger set of data reported by Plomteux (1980, Clin. Chem. 26 , 1897- 1899). Assuming these data were sampled from a bivariate normal population, evaluate the following quantities. (a) Compute the maximum likelihood estimates for the mean vector and the covariance matrix Σ . ) , ( 2 1 ' ~ µ µ µ = ~ X ⎡⎤ ⎢⎥ = ⎢⎥ ⎢⎥ ⎣⎦ ^ ∑= (b) Compute the unbiased estimate of the covariance matrix = S 1
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(c) The maximum likelihood estimate of the correlation between X 1 and X 2 and the t- statistic for testing H 0 : ρ = 0 are r = __________ t = _____________ df = ____________ (d) Use the Fisher z-transformation to construct an approximate 95% confidence interval for ρ . (e) The generalized sample variance is S = __________. (f) The total sample variance is ____________. (g) Use the Fisher z-transformation to construct an approximate 95% confidence interval for the correlation between X 1 and X 3 . lower limit = ______________ upper limit = ______________ (h) Use the Fisher z-transformation to construct an approximate 95% confidence interval for ρ 13 2 lower limit = ______________ upper limit = ______________ (i) In one sentence, how would you interpret the estimate of ρ 13 2 for these data? (j) Test the null hypothesis H 0 : ρ 13 2 = 0 against the alternative H A : ρ 13 2 > 0. Report t = __________ d.f. = __________ p-value = __________ State your conclusion. 2. Consider a random sample n ~ 2 ~ l ~ X , . . . , X , X from a p-dimensional normal population with mean vector ~ µ and covariance matrix Σ . The purpose of the problem is to construct the likelihood ratio test of the null hypothesis that the p attributes (or components of X) have the same variance σ 2 and are independent, that is, Σ = σ 2 I p × p . (a) Write down the formula for the natural logarithm of the joint likelihood function, when the null hypothesis is true, by substituting σ 2 I for = ) , ( 2 ~ σ µ A (b) Give formulas for the m. A .e.'s for ~ µ and σ 2 . 2
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