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stats134-4

stats134-4 - Stat134 Lec 3 Midterm I Solutions Problem 1(a...

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Stat134, Lec 3: Midterm I Solutions Problem 1: (a) Let X be the number of people who prefer Candidate A. P (530 X 570) = 570 X k =530 ± 1000 k ² p k (1 - p ) 1000 - k (b)Using the Normal Approximation Method, μ = 1000 p and σ = p 1000 p (1 - p ). P (530 X 570) Φ( 570 + . 5 - 1000 p p 1000 p (1 - p ) ) - Φ( 530 - . 5 - 1000 p p 1000 p (1 - p ) ) To ﬁnd the maximum point, take the derivative of the above expression and let it be 0: φ ( 570 + . 5 - 1000 p p 1000 p (1 - p ) ) h 141 p - 570 . 5 2 1000( p (1 - p )) 3 / 2 i - φ ( 530 - . 5 - 1000 p p 1000 p (1 - p ) ) h 59 p - 529 . 5 2 1000( p (1 - p )) 3 / 2 i = 0 exp {- 410 10(0 . 55 - p ) p (1 - p ) } = 59 p - 529 . 5 141 p - 570 . 5 p 0 . 5501 # Although it’s computationally hard, the idea is simple. (c) Since Φ( - 2 , 2) 95%, the largest “reasonable” p is corresponding to the point such that 570+ . 5 - 1000 p 1000 p (1 - p ) = - 2 and the smallest “reasonable” p is corresponding to the point such that

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stats134-4 - Stat134 Lec 3 Midterm I Solutions Problem 1(a...

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