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ESE 302 ADDITIONAL PRACTICE QUESTIONS FOR EXAM 1 Spring, 2011 1. Consider the problem of predicting gasoline consumption, Y , as a function of miles traveled, x . Here it is reasonable to assume that zero miles traveled involves zero gas consumption. So for a given set of observations,( , ), 1,. .., ii x yi n , it is natural to fit this relation in terms of a no-intercept regression model of the form: (1) i Yx   , n i ,..., 1 with iid errors, 2 ~( 0 ,) i N  , . ,..., 1 n i Given this model: ( a ) Determine the least-squares estimator, ˆ , of . ( b ) Show that ˆ is an unbiased estimator of . ( c ) Determine the variance of ˆ (Show all work). 2. A beer manufacturer is considering automating the final labeling and crating stages of his bottling plant. Before making a decision, he wishes to estimate the probability, p , that a randomly sampled bottle will be broken during this automated process. (i) If the Bernoulli random variables 1 B and 2 B denote the events that a randomly sampled bottle breaks during the labeling stage and crating stage, respectively, and if 11 Pr( 1) pB and 22 1 Pr( 1| 0) B  , then show that probability, p , is given by: (1) 121 2 ppp p p  (ii) A published study of automated labeling reported that 1 X of 1 N sampled bottles were broken, and a similar study of automated crating reported that 2 X of 2 N sampled bottles were broken. Assuming that these studies were independent, show that an unbiased estimator of p is given by (2) 2 2 ˆ X XX X p NNN N  (iii) Now suppose that a single study of the combined process was carried out using N bottles. In this case, 1 N above is simply the number of bottles,
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N , starting the process, and 2 N is the number of bottles “surviving” the labeling stage, i.e., (3) 21 NN X In terms of the above Bernoulli random variables, it must also be true that 2 Pr( 1) Pr( 0, BB B  [since a bottle can only be broken in crating if it “survives” the labeling stage]. Under these conditions, verify that 2 Pr( (1 ) B pp  , and use this to determine the bias (or unbiasedness) if the estimate in (2). 3. A US manufacturer of combustion engines is considering a new type of piston rod produced in England. Before deciding, they need to determine whether these rods are strong enough to function properly at the high cylinder temperatures of their engines. A British research firm has done a study of this type by regressing the stress-failure loads 1 ( ,. ., ) n y y of the new rods at a range of cylinder (piston crown) temperature settings 1 ( ,. ., ) n tt . Unfortunately their published beta estimates, 0 ˆ 5500 eng b = and 1 ˆ 10.2 eng b =- , are not directly applicable, since the stress-failure loads were measured in kilograms rather than pounds , and cylinder temperatures were measured in Centigrade rather than Fahrenheit . By obtaining the British data and performing the appropriate units
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  • Spring '08
  • ese302
  • unbiased estimator, Bernoulli random variables, randomly sampled bottle, cylinder temperatures, British research firm

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