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Unformatted text preview: AMS572.01 Midterm Exam Fall, 2009 ♠♣♥♦ Name: ________________________________ ID: _____________________ Signature: _________________________ Instruction: This is a close book exam. Anyone who cheats in the exam shall receive a grade of F. Please provide complete solutions for full credit. The exam goes from 12:502:10pm. Good luck! 1. (for all) To study the effectiveness of wall insulation in saving energy for home heating, the energy consumption (in MWh) for 5 houses in Bristol, England, was recorded for two winters; the first winter was before insulation and the second winter was after insulation: House 1 2 3 4 5 Before 12.1 10.6 13.4 13.8 15.5 After 12.0 11.0 14.1 11.2 15.3 (a) Please provide a 95% confidence interval for the difference between the mean energy consumption before and after the wall insulation is installed. What assumptions are necessary for your inference? (b) Can you conclude that there is a difference in mean energy consumption before and after the wall insulation is installed at the significance level 0.05? Please test it and evaluate the pvalue of your test. What assumptions are necessary for your inference? (c) Please write the SAS program to perform the test and examine the necessary assumptions given in (b). SOLUTION: This is inference on two population means, paired samples. (a). 36 . = d , s d = 1.30 CI: ) 97 . 1 , 25 . 1 ( 5 30 . 1 776 . 2 36 . = ⋅ ± (b). : = d H μ , : ≠ d H μ (1) 0.36 0 0.619 1.30/ 5 d d t s n = = ≈ 1, /2 4,0.025 2.776 n t t α = = Since 0.619 t ≈ is smaller than 4,0.025 2.776 t = , we cannot reject H . (2) 2 ( 0.619) 0.57 p value P T = ⋅ ≥ ≈ In the exam, since you do not have access to the statistical software such as R, you can only estimate the range of the pvalue based on your ttable. You thus estimate pvalue > 2*0.1=0.2 based on the ttable you were given. Assumptions for (a) and (b): the paired differences follow a normal distribution. (c) The SAS program is: Data energy; Input before after @@; Diff=before  after; Datalines; 12.1 12.0 10.6 11.0 13.4 14.1 13.8 11.2 15.5 15.3 ; Run; Proc univariate data = energy normal; Var Diff; Run; 2A (for AMS students). Suppose we have two independent random samples from two normal populations: ( 29 1 2 1 2 1 , , , ~ , n X X X N μ σ K , and ( 29 2 2 1 2 2 , ,...
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 Fall '10
 WeiZhu
 Normal Distribution, significance level, federal agency, random samples, Pivotal quantity, mean energy consumption

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