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PRACTICE PROBLEMS FOR TEST 1 MATH/STAT 352 (FALL 2005) Instructions • You need to show work to receive credit for non multiple-choice questions. • You are allowed to use calculator and one page formula sheet. 1. (2 points each) Consider an experiment of rolling a single die. Let A be the event that the number on a die is odd, and let B be the event that the number is less than or equal to 4. (a) Are A and B mutually exclusive (disjoint)? (A) YES (B) NO (b) Find the probability P (A ∪ B). (A) 26 (B) 36 (C) 46 (D) 56 (E) None of these (c) Find the probability P (A ∩ B). (A) 26 (B) 36 (C) 46 (D) 56 (E) None of these (d) Find the conditional probability P (A|B). (A) 26 (B) 36 (C) 46 (D) 56 (E) None of these (e) Are A and B independent? (A) YES (B) NO 2. (2 points each) The following data represent exam scores of five students: 86 76 89 89 90. (a) What is the mean score (sample mean)? (A) 76 (B) 86 (C) 89 (D) 90 (E) None of these (b) What is the median score? (A) 76 (B) 86 (C) 89 (D) 90 (E) None of these (c) What is the mode? (A) 76 (B) 86 (C) 89 (E) None of these (D) 90 1 (d) What is the range of scores? (A) 3(B) 4 (C) 10 14 (D) (E) None of these (e) What is the sample variance of the scores? (A) 4 (B) 26.8 (C) 33.5 (D) 134 (E) None of these 3. (2 points each) An experimenter is studying the effects of temperature, pressure, and type of catalyst on yield from a certain chemical reaction. Three different temperatures, four different pressures, and five different catalysts are under consideration (a) If any particular experimental run involves the use of a single temperature, pressure, and catalyst, how many experimental runs are possible? (A) 3 (B) 12 (C) 60 (D) 220 (E) None of these (b) How many experimental runs are there that involve the use of the lowest temperature and two lowest pressures? (A) 2 (B) 3 (C) 8 (D) 10 (E) None of these 4. (2 points each) Consider a continuous random variable X with the density function −2x ke for x ≥ 0 f (x) = 0 otherwise (a) Determine the value of the constant k [a density must integrate to 1!] (b) Find the cumulative distribution function of X. (c) What is the probability P (2 < X < 10)? (d) Find the mean of X. (e) Find the median of X. 5. (5 points) A company that manufactures video cameras produces a basic model and a deluxe model. Over the past year, 40% of the cameras sold have been of the basic model. Of those buying the basic model, 30% purchase an extended warranty, wheres 50% of all deluxe purchasers do so. If you learn that a randomly selected purchaser has an extended warranty, how likely is it that he or she has a basic model? 2 6. (5 points) A boiler has five identical relief valves. The probability that any particular valve will open on demand is 0.95. Assuming independent operation of the valves, Calculate the probability that at least one valve opens on demand. 7. (2 points each) A chemical supply company currently has in stock 100 lb of a certain chemical. which it sells to customers in 5-lb lots. Let X be the number of lots ordered by a randomly chosen customer, and suppose that the pmf of X is x p(x) 1 2 3 4 0.2 0.4 0.3 0.1 (a) If F is the distribution function of X, then what is F (2)? (b) What is the probability that a customer order fewer than 2 lots? (c) Compute the expected value of X. (d) Compute the variance of X. 8. (2 points each) An urn contains 5 white and 15 red balls. If 10 balls are randomly selected from the urn without replacement, find the probability that exactly 3 white are obtained. 9. (5 points) Let X be the damage incurred (in $) in a certain type of accident during a given year. Possible X values are 0, 1000, 5000, and 10000, with probabilities 0.8, 0.1, 0.08, and 0.02, respectively. A particular company offers a $500 deductible policy. If the company wishes its expected profit to be $100, what premium amount should it charge? 3