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Unformatted text preview: AMS572.01 Final Exam Fall, 2007 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 5:00-7:30pm. 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 . = , and conclude that the average corneal thicknesses are unaffected by glaucoma. (b) A 90% CI for 2 1 - is given by ] 198 . 3 , 198 . 11 [ 8 744 . 10 895 . 1 4 2 , 1- = - = - n s t d d n (c) The SAS code is as follows. Data eyes; Input bad good @@; Diff=bad-good; 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; 2 (for all students in class). Let X i , i = 1, , n, denote the outcome of a series of n independent trials, where X i = 1 with probability p, and Xi = 0 with probability (1- p). Let = = n i i X X 1 . (a). Please derive the 100(1-)% large sample confidence interval for p using the pivotal quantity method. (b). At the significance level , please derive the large sample test for H : p = p versus Ha: p p , using the pivotal quantity method. (* Please include the derivation of the pivotal quantity, the proof of its distribution, and the derivation of the rejection region for full credit.) Solution: (a). The population distribution is Bernoulli (p), i.e. X i ~ Bernoulli(p). Therefore the population mean is p and the population variance is p(1-p). When the sample size n is large, by the central limit theorem, we know that the sample mean follows approximately the normal distribution with its mean being the population mean and its variance being the population variance divided by n as follows: ( 29 - = = = n p p p N n X n X p n i i 1 , ~ 1 . Thus it is easily shown that ( 29 ( 29 1 , ~ 1 N n p p p p Z -- = is a pivotal quantity for the inference on p....
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