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Unformatted text preview: Problems and Solutions for Stat/ME 424, Spring 2011 First Midterm Examination (Total : 100 points) This is a closed book exam, except for 23 sides of notes. You are allowed to use calculators. There are four problems in this exam. In order to get full credit, please answer all problems. Please put your answers in a blue book with your name on the blue book cover. Problem 1 (30 pts) A simple linear regression model y = + 1 x + with N ( , 2 ) is fitted to the accom panying data on x = % refusederived fuel (RDF) heat input and y = % efficiency for a certain boiler. x 37 30 48 29 27 16 20 y 78.0 77.2 74.4 77.7 76.9 79.0 82.1 76.5 with y = 77 . 725, x = 25 . 875, = 81 . 173, ( x i x ) 2 = 1442 . 875 and RSS = 8 . 7328 (a) Compute a 95% confidence interval for 1 . You may find the following information useful. Upper percentiles of t distributions: t 6 , . 025 = 2 . 45 , t 6 , . 05 = 1 . 94 , t 7 , . 05 = 1 . 89 , t 7 , . 025 = 2 . 36 , t 8 , . 025 = 2 . 31 , t 8 , . 05 = 1 . 86 . Since = y 1 x , 1 = y x = 77 . 725 81 . 173 25 . 875 = . 133. Also 2 = 8 . 7328 6 = 1 . 455 and = 1 . 206. The 95% CI for 1 is . 133 t 6 , . 025 ( x i x ) 2 = . 133 2 . 45 1 . 206 1442 . 875 = [ . 211 , . 055 ] . (b) Test H : 1 = 0 versus H a : 1 6 = 0 using the significant level = . 05. That is, with justification answer whether H should be rejected at = . 05 or not. Since the CI in (a) does not contain 0, H : 1 = 0 should be rejected. (c) In addition to the boiler considered above, there is a second boiler where x = % refuse derived fuel (RDF) heat input and y = % efficiency are related via the simple linear regression model y = + 1 x + with N ( , 2 ) . Note that both boilers have the same error variance 1 2 . If the data set consists of n 1 observations on the first boiler and n 2 on the second and if RSS 1 and RSS 2 denote the two residual sums of squares, then a pooled estimate of...
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This note was uploaded on 04/26/2011 for the course STATS 424 taught by Professor Peterqian during the Spring '11 term at Wisconsin.
 Spring '11
 PETERQIAN

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