P 45 X 50 P X 45 P X 46 P X 50 P X x C n x p x1 p n x n x n x p x 1 p n x P X

# P 45 x 50 p x 45 p x 46 p x 50 p x x c n x p x1 p n x

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P (45 X 50) = P ( X = 45) + P ( X = 46) + · · · + P ( X = 50) P ( X = x ) = C n x p x (1 p ) n x = n ! x !( n x )! p x (1 p ) n x P ( X = 45) = C 100 45 (0 . 4) 45 (0 . 6) 55 = 100! 45! × 55! (0 . 4) 45 (0 . 6) 55 P ( X = 46) , P ( X = 47) , P ( X = 48) , P ( X = 49) , P ( X = 50) 13
Section 5.4: Normal Distribution Approximation for Binomial Distribution X Binomial( n, p ), E ( X ) = np , V ar ( X ) = np (1 p ) Result : If the number of trial n is large — such that np (1 p ) > 5 — then the distribution of the random variable Z = X E ( X ) V ar ( X ) = X np np (1 p ) is approximately a standard normal distribution. Z = X np np (1 p ) approx N (0 , 1) 14
Z = X np np (1 p ) approx N (0 , 1) Example 5.8 Customer Visits Generated from Web Page Contacts, p. 202 Mary David makes the initial telephone contact with customers who have responded to an advisement on her company’s Web Page in an e ff ort to assess whether a follow-up visit to their hoomes is likely to be worthwhile. Her experience suggests that 40% of the initial contacts lead to follow-up visits. If she has 100 Web page contacts, what is the probability that between 45 and 50 home visits will result? 15

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