Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Stat312: Sample Final Exam Moo K. Chung, Yulin Zhang [email protected], [email protected] May 16, 2003 1. Let X 1 , · · · , X n be a random sample from Bernoulli distribution with parameter p . (a) Find the maximum likelihood estimator of (1 - p ) 2 (20pts). (c) Is your estimator in (a) unbiased? If it is biased, find an unbiased estimator of (1 - p ) 2 (20pts). Solution. (a) First find the MLE for p . Note that the Bernoulli probability function is P ( X i = x ) = p x (1 - p ) 1 - x for x = 0 , 1 (5pts). The likelihood function is then L ( x 1 , · · · , x n ; p ) = p n i =1 x i (1 - p ) n i =1 1 - x i (5pts). By solving log L/∂p = 0 , we get ˆ p = ¯ x (5pts). From the invariance principle, \ (1 - p ) 2 = (1 - ¯ x ) 2 (5pts). (b) E (1 - ¯ X ) 2 = 1 - 2 E ¯ X + E ¯ X 2 . Note that E ¯ X 2 = Var ¯ X - ( E ¯ X ) 2 = p (1 - p ) /n - p 2 . So E (1 - ¯ X ) 2 = 1 - 2 p + p (1 - p ) /n - p 2 . Hence it is biased (5pts). We only need to find an unbiased estimator of p 2 . See Midterm I, where it is given by ¯ X - S 2 , where S is the sample variance. So the unbiased estimator of (1 - p ) 2 is 1 - 2 ¯ X + ¯ X - S 2 = 1 - ¯ X - S 2 (15pts). 2. Gross sales before and after a training program is given by Salesperson : 1 2 3 4 5 6 Sales before: 90 83 105 97 110 78 Sales after : 97 80 110 93 123 84 Determine if the training program is effective at α = 0 . 1 . Carefully do your analysis stating all the relevant assumptions (30 pts). Solution. This problem can be also solved using regression analysis. The points will be given in three categories. Model specification (10 pts). We assume that the gross sale X i of i -th sales person before the training to follow normal, i.e. X i N ( μ b , σ 2 b ) . We also assume that the gross sale
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