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Unformatted text preview: Homework 1 Solutions 1. The Binomial Test H : . 5 = 70 H a : . 5 > 70 , Under H , Pr ( X 70) = Pr ( X > 70) = 0 . 5 . Thus, hypothesis can be rewritten as H : p = 0 . 5 H a : p > . 5 . Test Statistics: b = n summationdisplay i =1 I ( x i > 70) = 38 Normal Approximation: z * = B . 5 * n . 25 * n = 5 . 69 p value = Pr ( Z > z * ) = 1 (5 . 69) , where Z N (0 , 1). The exact pvalue: p value = Pr ( B 38) = 40 summationdisplay b =38 parenleftbigg 40 b parenrightbigg (0 . 5) 40 . Therefore, we conclude that we reject H , thus there is statistical evidence that the median score has increased after the course content was revised. 2. Confidence Intervals (a) a 90% CI for the median: Pr ( X ( a ) < . 5 < X ( b ) ) 1 = 0 . 9 From the normal approximation and the symmetricity of binomial distri bution, we can get a = 15 and b = 26. Therefore, a 90% approximate confidence interval is given by ( X ( a ) , X ( b ) ) = ( X (15) , X (26) ) = (76 , 80) . The achieved confidence level: Pr ( X (15) . 5 < X (26) ) = 25 summationdisplay b =15 parenleftbigg 40 b parenrightbigg (0 . 5) 40 . 9193 1 Lets consider ( X (14) , X (25) ) = (76 , 80) . The confidence intervals are the same as above, but the achieved confidence level is Pr ( X (14) . 5 < X (25) ) = 24 summationdisplay b =14 parenleftbigg 40 b parenrightbigg (0 . 5) 40 . 9038 ....
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This note was uploaded on 07/22/2011 for the course STA 4502 taught by Professor Staff during the Spring '08 term at University of Florida.
 Spring '08
 Staff
 Statistics, Binomial

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