Stat 511 Final Exam S2009

Stat 511 Final Exam S2009 - Stat 511 Final Exam May 4 2009...

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1 Stat 511 Final Exam May 4, 2009 Prof. Vardeman ( This exam will scored on a 160 point basis. ) I have neither given nor received unauthorized assistance on this exam. ________________________________________________________ Name _______________________________________________________________ Name Printed
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2 1. A marketing study used as an example in Neter et al. concerned counts of customers visiting a particular lumber store during a two-week period from each of 110 n = different census tracts (these are metropolitan areas with populations of about 4000 residents each). Various demographic characteristics of the tracts were also obtained. Available for each tract are 1 2 3 the number of customers visiting the store from the tract the number of housing units in the tract the average personal income in the tract (dollars/year) the average housing unit age in the t y x x x = = = = 4 5 ract (years) the distance from the tract to the nearest competing store (miles) the distance from the tract to the store (miles) x x = = Here we will model customer counts as independent Poisson variables with E ii y λ = and 0 1 12 23 34 45 5 ln i i i i x xxxx ββ β =+ + + + + There is an R output for these data attached to this exam. Use it to help you answer the following. a) In the presence of all other predictors, which of the 's x appears to be least important to the description of y ? Explain . b) What are approximate 95% confidence limits for the log-mean number of visits by tract #1 customers in a two week period? What are corresponding approximate 95% limits for the mean number of such visits? 8 pts 8 pts
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3 c) Find an appropriate prediction standard error for the number of visits in a future two week period by tract #1 customers. (Hints: n 01 12 23 34 45 5 ln bb xb x λ = + ++++ . How is a standard error for m () ˆ exp ln = related to a standard error for n ln ? How is Var new Y related to ?) d) Census tract #41 has one of the largest values of y in the data set (and corresponding large value of ˆ ) and thus is an important source of customers for the store. A competitor is about to open a new store only .1 mile away from this tract. By what fraction does the fitted model suggest that the mean number of visits from tract #41 customers will decrease? Provide 95% confidence limits for this fraction. 10 pts 8 pts
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4 2. Consider the making of fitted values ˆ i y from n pairs ( ) , ii x y under a model that says ( ) i yx μ ε = + for some unknown mean function () x where the i are iid with mean 0 and variance 2 σ . A standard measure of the flexibility of the fitting method employed is 2 1 1 ˆ Cov , n i flex y y = = For "linear" fitting methods (ones for which ˆ = YM Y for some fixed nn × matrix M ) this is fairly easily computed. (Consider the covariance matrix of the 21 n × vector ˆ , YY computed beginning from 2 Var = YI .) a) What is numerical value of flex for simple linear regression fitting? (Note that here, = X MP , the projection matrix onto the column space of the simple linear regression X matrix.) Explain .
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Stat 511 Final Exam S2009 - Stat 511 Final Exam May 4 2009...

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