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CHAPTER 8 SECTION 1: CONTINUOUS PROBABILITY DISTRIBUTIONS MULTIPLE CHOICE 1. Which of the following represents a difference between continuous and discrete random variables? a. Continuous random variables assume an uncountable number of values, and discrete random variables do not. b. The probability for any individual value of a continuous random variable is zero, but for discrete random variables it is not. c. Probability for continuous random variables means finding the area under a curve, while for discrete random variables it means summing individual probabilities. d. All of these choices are true. ANS: D PTS: 1 REF: SECTION 8.1 2. Which of the following is always true for all probability density functions of continuous random variables? a. The probability at any single point is zero. b. They contain an uncountable number of possible values. c. The total area under the density function f(x) equals 1. d. All of these choices are true. ANS: D PTS: 1 REF: SECTION 8.1 3. Suppose f(x) = 0.25. What range of possible values can X take on and still have the density function be legitimate? a. [0, 4] b. [4, 8] c. [2, +2] d. All of these choices are true. ANS: D PTS: 1 REF: SECTION 8.1 4. The probability density function, f(x), for any continuous random variable X, represents: a. all possible values that X will assume within some interval a x b. b. the probability that X takes on a specific value x. c. the height of the density function at x. d. None of these choices. ANS: C PTS: 1 REF: SECTION 8.1 5. Which of the following is true about f(x) when X has a uniform distribution over the interval [a, b]? a. The values of f(x) are different for various values of the random variable X. b. f(x) equals one for each possible value of X. c. f(x) equals one divided by the length of the interval from a to b. d. None of these choices. ANS: C PTS: 1 REF: SECTION 8.1 6. The probability density function f(x) for a uniform random variable X defined over the interval [2, 10] is a. 0.125 b. 8 c. 6 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. d. None of these choices. ANS: A PTS: 1 REF: SECTION 8.1 7. If the random variable X has a uniform distribution between 40 and 50, then P(35 X 45) is: a. 1.0 b. 0.5 c. 0.1 d. undefined. ANS: B PTS: 1 REF: SECTION 8.1 8. The probability density function f(x) of a random variable X that has a uniform distribution between a and b is a. (b + a)/2 b. 1/b 1/a c. (a b)/2 d. None of these choices. ANS: D PTS: 1 REF: SECTION 8.1 9. Which of the following does not represent a continuous uniform random variable? a. f(x) = 1/2 for x between 1 and 1, inclusive. b. f(x) = 10 for x between 0 and 1/10, inclusive. c. f(x) = 1/3 for x = 4, 5, 6. d. None of these choices represents a continuous uniform random variable. ANS: C PTS: 1 REF: SECTION 8.1 10. Suppose f(x) = 1/4 over the range a x b, and suppose P(X > 4) = 1/2. What are the values for a and b? a. 0 and 4 b. 2 and 6 c. Can be any range of x values whose length (b a) equals 4. d. Cannot answer with the information given. ANS: B PTS: 1 REF: SECTION 8.1 11. What is the shape of the probability density function for a uniform random variable on the interval [a, b]? a. A rectangle whose X values go from a to b. b. A straight line whose height is 1/(b a) over the range [a, b]. c. A continuous probability density function with the same value of f(x) from a to b. d. All of these choices are true. ANS: D PTS: 1 REF: SECTION 8.1 TRUE/FALSE 12. A continuous probability distribution represents a random variable having an infinite number of outcomes which may assume any number of values within an interval. ANS: T PTS: 1 REF: SECTION 8.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 13. Continuous probability distributions describe probabilities associated with random variables that are able to assume any finite number of values along an interval. ANS: F PTS: 1 REF: SECTION 8.1 14. A continuous random variable is one that can assume an uncountable number of values. ANS: T PTS: 1 REF: SECTION 8.1 15. Since there is an infinite number of values a continuous random variable can assume, the probability of each individual value is virtually 0. ANS: T PTS: 1 REF: SECTION 8.1 16. A continuous random variable X has a uniform distribution between 10 and 20 (inclusive), then the probability that X falls between 12 and 15 is 0.30. ANS: T PTS: 1 REF: SECTION 8.1 17. A continuous random variable X has a uniform distribution between 5 and 15 (inclusive), then the probability that X falls between 10 and 20 is 1.0. ANS: F PTS: 1 REF: SECTION 8.1 18. A continuous random variable X has a uniform distribution between 5 and 25 (inclusive), then P(X = 15) = 0.05. ANS: F PTS: 1 REF: SECTION 8.1 19. We distinguish between discrete and continuous random variables by noting whether the number of possible values is countable or uncountable. ANS: T PTS: 1 REF: SECTION 8.1 20. In practice, we frequently use a continuous distribution to approximate a discrete one when the number of values the variable can assume is countable but very large. ANS: T PTS: 1 REF: SECTION 8.1 21. Let X represent weekly income expressed in dollars. Since there is no set upper limit, we cannot identify (and thus cannot count) all the possible values. Consequently, weekly income is regarded as a continuous random variable. ANS: T PTS: 1 REF: SECTION 8.1 22. To be a legitimate probability density function, all possible values of f(x) must be non-negative. ANS: T PTS: 1 REF: SECTION 8.1 23. To be a legitimate probability density function, all possible values of f(x) must lie between 0 and 1 (inclusive). ANS: F PTS: 1 REF: SECTION 8.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 24. The sum of all values of f(x) over the range of [a, b] must equal one. ANS: F PTS: 1 REF: SECTION 8.1 25. A probability density function shows the probability for each value of X. ANS: F PTS: 1 REF: SECTION 8.1 26. If X is a continuous random variable on the interval [0, 10], then P(X > 5) = P(X 5). ANS: T PTS: 1 REF: SECTION 8.1 27. If X is a continuous random variable on the interval [0, 10], then P(X = 5) = f(5) = 1/10. ANS: F PTS: 1 REF: SECTION 8.1 28. If a point y lies outside the range of the possible values of a random variable X, then f(y) must equal zero. ANS: T PTS: 1 REF: SECTION 8.1 COMPLETION 29. A(n) ____________________ random variable is one that assumes an uncountable number of possible values. ANS: continuous PTS: 1 REF: SECTION 8.1 30. For a continuous random variable, the probability for each individual value of X is ____________________. ANS: zero 0 PTS: 1 REF: SECTION 8.1 31. Probability for continuous random variables is found by finding the ____________________ under a curve. ANS: area PTS: 1 REF: SECTION 8.1 32. A(n) ____________________ random variable has a density function that looks like a rectangle and you can use areas of a rectangle to find probabilities for it. ANS: uniform PTS: 1 REF: SECTION 8.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 33. Suppose X is a continuous random variable for X between a and b. Then its probability ____________________ function must non-negative for all values of X between a and b. ANS: density PTS: 1 REF: SECTION 8.1 34. The total area under f(x) for a continuous random variable must equal ____________________. ANS: 1 one PTS: 1 REF: SECTION 8.1 35. The probability density function of a uniform random variable on the interval [0, 5] must be ____________________ for 0 x 5. ANS: 1/5 0.20 PTS: 1 REF: SECTION 8.1 36. To find the probability for a uniform random variable you take the ____________________ times the ____________________ of its corresponding rectangle. ANS: base; height height; base length; width width; length PTS: 1 REF: SECTION 8.1 37. You can use a continuous random variable to ____________________ a discrete random variable that takes on a countable, but very large, number of possible values. ANS: approximate PTS: 1 REF: SECTION 8.1 SHORT ANSWER 38. A continuous random variable X has the following probability density function: f(x) = 1/4, 0 x 4 Find the following probabilities: a. P(X 1) b. P(X 2) c. P(1 X 2) d. P(X = 3) This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. ANS: a. 0.25 b. 0.50 c. 0.25 d. 0 PTS: 1 REF: SECTION 8.1 Waiting Time The length of time patients must wait to see a doctor at an emergency room in a large hospital has a uniform distribution between 40 minutes and 3 hours. 39. {Waiting Time Narrative} What is the probability density function for this uniform distribution? ANS: f(x) = 1/140, 40 x 180 (minutes) PTS: 1 REF: SECTION 8.1 40. {Waiting Time Narrative} What is the probability that a patient would have to wait between one and two hours? ANS: 0.43 PTS: 1 REF: SECTION 8.1 41. {Waiting Time Narrative} What is the probability that a patient would have to wait exactly one hour? ANS: 0 PTS: 1 REF: SECTION 8.1 42. {Waiting Time Narrative} What is the probability that a patient would have to wait no more than one hour? ANS: 0.143 PTS: 1 REF: SECTION 8.1 43. The time required to complete a particular assembly operation has a uniform distribution between 25 and 50 minutes. a. What is the probability density function for this uniform distribution? b. What is the probability that the assembly operation will require more than 40 minutes to complete? c. Suppose more time was allowed to complete the operation, and the values of X were extended to the range from 25 to 60 minutes. What would f(x) be in this case? This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. ANS: a. f(x) = 1/25, 25 x 50 b. 0.40 c. f(x) = 1/35, 25 x 60 PTS: 1 REF: SECTION 8.1 44. Suppose f(x) equals 1/50 on the interval [0, 50]. a. What is the distribution of X? b. What does the graph of f(x) look like? c. Find P(X 25) d. Find P(X 25) e. Find P(X = 25) f. Find P(0 < X < 3) g. Find P(3 < X < 0) h. Find P(0 < X < 50) ANS: a. X has a uniform distribution on the interval [0, 50]. b. f(x) forms a rectangle of height 1/50 from x = 0 to x = 50. c. 0.50 d. 0.50 e. 0 f. 0.06 g. 0.06 h. 1.00 PTS: 1 REF: SECTION 8.1 Chemistry Test The time it takes a student to finish a chemistry test has a uniform distribution between 50 and 70 minutes. 45. {Chemistry Test Narrative} What is the probability density function for this uniform distribution? ANS: f(x) = 1/20, 50 x 70 PTS: 1 REF: SECTION 8.1 46. {Chemistry Test Narrative} Find the probability that a student will take more than 60 minutes to finish the test. ANS: 0.50 PTS: 1 REF: SECTION 8.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 47. {Chemistry Test Narrative} Find the probability that a student will take no less than 55 minutes to finish the test. ANS: 0.75 PTS: 1 REF: SECTION 8.1 48. {Chemistry Test Narrative} Find the probability that a student will take exactly one hour to finish the test. ANS: 0 PTS: 1 REF: SECTION 8.1 49. {Chemistry Test Narrative} What is the median amount of time it takes a student to finish the test? ANS: 60 minutes PTS: 1 REF: SECTION 8.1 50. {Chemistry Test Narrative} What is the mean amount of time it takes a student to finish the test? ANS: 60 minutes PTS: 1 REF: SECTION 8.1 Elevator Waiting Time In a shopping mall the waiting time for an elevator is found to be uniformly distributed between 1 and 5 minutes. 51. {Elevator Waiting Time Narrative} What is the probability density function for this uniform distribution? ANS: f(x) = 1/4, 1 x 5 PTS: 1 REF: SECTION 8.1 52. {Elevator Waiting Time Narrative} What is the probability of waiting no more than 3 minutes? ANS: 0.50 PTS: 1 REF: SECTION 8.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher. 53. {Elevator Waiting Time Narrative} What is the probability that the elevator arrives in the first minute and a half? ANS: 0.125 PTS: 1 REF: SECTION 8.1 54. {Elevator Waiting Time Narrative} What is the median waiting time for this elevator? ANS: 3 minutes PTS: 1 REF: SECTION 8.1 This edition is intended for use outside of the U.S. only, with content that may be different from the U.S. Edition. This may not be resold, copied, or distributed without the prior consent of the publisher.