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Unformatted text preview: AMS572.01 Practice 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 3:505:10pm. Good luck! 1. (for all students in class) To determine whether glaucoma affects the corneal thickness, measurements were made in 8 people affected by glaucoma in one eye but not in the other. The corneal thickness (in microns) were as follows: Person 1 2 3 4 5 6 7 8 Eye affected 488 478 480 426 440 410 458 460 Eye not affected 484 478 492 444 436 398 464 476 (a) According to the data, can you conclude, at the significance level of 0.10, that the corneal thickness is not equal for affected versus unaffected eyes? (b) Calculate a 90% confidence interval for the mean difference in thickness. (c) Please write the entire SAS code to check the assumptions necessary in (a) and to perform the test asked for in (a). Solution: (a) Using 4 = d and 744 . 10 = d s , the test statistic is 053 . 1 8 744 . 10 4 = = = n s d t d Since 895 . 1 05 . , 1 8 = t t s , do not reject H at 10 . = α , and conclude that the average corneal thicknesses are unaffected by glaucoma. (b) A 90% CI for 2 1 μ μ is given by ] 198 . 3 , 198 . 11 [ 8 744 . 10 895 . 1 4 2 , 1 = × ± = ⋅ ± n s t d d n α (c) The SAS code is as follows. Data eyes; Input bad good @@; Diff=badgood; Datalines; 488 484 478 478 480 492 426 444 440 436 410 398 458 464 460 476 ; Run; Proc univariate data = eyes normal; Var diff; Run; 2A (for AMS PhD students). Suppose we have two independent random samples from two normal populations: ( 29 2 1 2 1 , ~ , , , 1 σ μ N X X X n , and ( 29 2 2 2 1 , ~ , , , 2 σ μ N Y Y Y n . Please derive the pooledvariance ttest using the pivotal quantity method. Please make sure that you include the following key steps. (a) Please derive the distribution of ( 29 Y X (b) Please derive the distribution of ( 29 ( 29 [ ] 2 2 2 2 2 1 1 / 1 1 σ S n S n + (c) Please derive the distribution of the pooledvariance t statistic (the pivotal quantity). (d) Please derive the rejection region for a 2sided test at the significance level of α. (e) Please illustrate using the pdf plot how to calculate the pvalue for a 2sided test. Solution: ( Please refer to your lecture notes for the entire derivation.) Here is a simple outline of the derivation....
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 Fall '10
 WeiZhu
 Normal Distribution, Variance, Pivotal quantity, George Williams, John Pauzke, AMS PhD

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