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# hw4sol - Stat 511 Homework 4 Solution Spring 2011 up...

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Stat 511 Homework 4 Solution Spring 2011 1. Last HW had the β vector and X matrix for a very messed up experiment. They were: μ α 1 α 2 α 3 α 4 α 5 α 6 1 1 0 1 1 0 0 1 1 0 0 1 0 0 1 1 1 0 0 1 1 1 1 1 1 0 1 1 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 1 1 1 0 0 1 0 1 1 1 0 0 1 0 1 1 The X matrix and data, y , are in the file ’messedup.txt’ on the class web site. (a) Estimate a 90% confidence interval for α 4 ( α 2 + α 5 + α 6 ). The corresponding C is C = (0 , 0 , 1 , 0 , 1 , 1 , 1), thus the 90% C.I is ± t 3 , 95% σ 2 C ( X X ) C = [0 . 7896 , 2 . 9104] (b) Test Ho: α 1 = 0. (N.B. α 1 is estimable). Report your test statistic and p-value. The corresponding C is C = (0 , 1 , 0 , 0 , 0 , 0 , 0), thus the test statistic T is T = d σ 2 C ( X X ) C = 1 . 7259 , thus the p-value is P ( | t 3 | > T ) = 0 . 1828, fail to reject the null hypothesis α 1 = 0. (c) Test Ho: α 1 = 0 , α 2 = 0 , α 4 = 0 , and α 5 + α 6 = 0. (N.B. This hypothesis is testable) Report your test statistic and p-value. The corresponding C is C = 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 with full rank 4 and the test statistic F is F = ( d ) ( C ( X X ) C ) 1 ( d ) / 4 σ 2 = 32 . 7214 The p-value is P ( F 4 , 3 > F ) = 0 . 008, then we reject the null hypothesis, namely, at least one of α 1 , α 2 , α 4 , and α 5 + α 6 is not zero. 2. This problem is based on a research proposal I read last week. A researcher on campus is studying whether monkeys are selective in what they eat. One of the major items in the diet 1

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Stat 511 Homework 4 Solution Spring 2011 of a particular group of monkeys is the fruit of the baobab tree. The researcher is measur- ing various nutrient characteristics of the baobab fruits. One of the most important is the sugar concentration. This is not easy to measure and even harder to measure precisely. The researcher wanted to estimate the precision of her method of determining baobab sugar con- centration. There are no standard reference materials (samples with known chararacteristics) for baobab fruits, so the measurement error has to be estimated from multiple measurements on the same fruits. The researcher collected 8 fruits and measured each twice. (Ask if you don’t remember how to estimate measurement variance from this sort of data). The data are in baobab.txt on the class web site.
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hw4sol - Stat 511 Homework 4 Solution Spring 2011 up...

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