STAT 400 Lecture AL1 Examples for 5.7 Spring 2015 Dalpiaz Normal Approximation to Binomial Distribution: Normal Binomial mean n p standard deviation 1 ppn Binomial, n = 25, p = 0.50. Normal, = 25 0.50 = 12.5, 2 = 25 0.50 0.50 = 6.25. = 2.5. Binomial, n = 25, p = 0.10. Normal, = 25 0.10 = 2.5, 2 = 25 0.10 0.90 = 2.25. = 1.5. 5 pn 5 1 pn 0 1 3 ppnpn nppnpn 1 3 1. A fair coin is tossed 25 times. Let X denote the number of H's. a) Find the probability P(X = 17). b) Use Normal approximation to find the probability P(X = 17). = n p = 1 ppn c) Find the probability P(X 11). d) Use Normal approximation to find the probability P(X 11). e) Find the probability P(10 X 14). f) Use Normal approximation to find the probability P(10 X 14). Binomial, n = 25, p = 0.50 PMF CDF 0 0.0000 0.0000 1 0.0000 0.0000 2 0.0000 0.0000 3 0.0001 0.0001 4 0.0004 0.0005 5 0.0016 0.0020 6 0.0053 0.0073 7 0.0143 0.0216 8 0.0322 0.0539 9 0.0609 0.1148 10 0.0974 0.2122 11 0.1328 0.3450 12 0.1550 0.5000 13 0.1550 0.6550 14 0.1328 0.7878 15 0.0974 0.8852 16 0.0609 0.9461 17 0.0322 0.9784 18 0.0143 0.9927 19 0.0053 0.9980 20 0.0016 0.9995 21 0.0004 0.9999 22 0.0001 1.0000 23 0.0000 1.0000 24 0.0000 1.0000 25 0.0000 1.0000 2. An airline knows that about 15% of the people who buy tickets for a certain flight cancel their reservations. The airline sells 100 tickets for a flight that contains only 92 seats. Assuming that each person either cancels the reservation or not independently, use Normal approximation to find the probability that there will be enough seats for all the passengers.3. A fair 6-sided die is rolled 180 times. a) Find the exact probability that “6” shows up exactly 35 times? b) Use Normal approximation to find the probability that “6” shows up exactly 35 times? c) Use Normal approximation to find the probability that “6” shows up at least 35 times? d) Use Normal approximation to find the probability that “6” shows up between 20 and 40 times (both inclusive)?Normal Approximation to Poisson Distribution: Normal Poisson mean standard deviation λ 4. Traffic accidents at a particular intersection follow Poisson distribution with an average rate of 1.4 per week. Use Normal approximation to find the following: a) Find the exact probability that there would be exactly 68 accidents at this intersection in one year (52 weeks). b) Use Normal approximation to find the probability that there would be exactly 68 accidents at this intersection in one year. c) Use Normal approximation to find the probability that there would be at most 70 accidents at this intersection in one year. d) Use Normal approximation to find the probability that there would be between 65 and 80 (both inclusive) accidents at this intersection in one
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