Stat 401 Practice Final Exam

Stat 401 Practice Final Exam - STATISTICS 401 PRACTICE...

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STATISTICS 401: PRACTICE PROBLEMS FOR FINAL EXAM Spring 2000 1. The probability that an American engineering firm will establish a branch office in Toronto is .7, the probability that it will establish a branch office in Mexico City is .4 and the probability it will establish a branch office in at least one of the cities is .8. Define events A and B as follows: A = American engineering firm will establish a branch office in Toronto B = American engineering firm will establish a branch office in Mexico City It may be useful to draw Venn diagrams. For parts (a)-(c), what is the probability that a branch office will be established in (a) in both cities? (b) neither of the cities? (c) exactly one of the cities? (d) Are events A and B independent? Why or why not? Hint for problem 1 For any events A and B, P(A or B) = P(A) + P(B) - P(A and B). Events A and B are independent if and only if P(A and B) = P(A)P(B). 2. From a load of 50 Panasonic Tape recorders 35 are destined for New York and 15 for Boston. If two are shipped to Trenton by mistake and the ``selection'' is random, what are the probabilities that a) both tape recorders should have gone to New York; b) one should have gone to New York and one to Boston. Hint for problem 2 The distribution of NY recorders shipped to Trenton is hypergeometric (see p. 129). 3. From 5 biologists, 4 chemists, and 3 physicists, (a) how many committees of size 4 can be formed? (b) how many committees containing 2 biologists, 1 chemist, and 1 physicist can be formed? Hint for problem 3
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The number of ways of selecting a group of k from a group of n is C(n,k) or n choose k. See discussion of combinations, p. 71. 4. Seventy percent of the light aircraft that disappear while in flight in a certain country are subsequently discovered. Of the aircraft that are discovered, 60 have an emergency locator, whereas 90 of the aircraft not discovered do not have such a locator. Define events A and B as follows: A = light aircraft that disappears is discovered B = light aircraft that disappears has an emergency locator Suppose a light aircraft has disappeared. (a) What is the probability that it has an emergency locator and it will not be discovered? (b) What is the probability that it has an emergency locator? (c) If it has an emergency locator, what is the probability that it will not be discovered? Hint for problem 4 You're given P(A), P(B|A), and P(B'|A'). Use these to find P(A'), P(B'|A), and P(B|A') then use Bayes' theorem. 5. Grafting, the uniting of the stem of one plant with the stem or root of another, is widely used commercially to grow the stem of one variety that produces fine fruit on the root system of another variety with a hardy root system. For example, most sweet oranges grow on trees grafted to the root of a sour orange variety. Suppose that each graft fails independently with probability .3. a) Write the formula for the p.m.f. for X , the number of grafts that fail in a series of five trials. b) Plot the cumulative distribution function F . c) Suppose that the cost of each failed graft is $9.00. Find (i) the probability that the cost from failed grafts will exceed $20.00 in the five trials.
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