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Homework answers / question archive / 1)A uniform continuous distribution is also referred to as a rectangular distribution

1)A uniform continuous distribution is also referred to as a rectangular distribution.

Ans: True

2. The height of the rectangle depicting a uniform distribution is the probability of each outcome and it same for all of the possible outcomes

Ans: False

3. The area of the rectangle depicting a uniform distribution is always equal to one.

Ans: True

4. Many human characteristics such as height and weight and many measurements such as variables such as household insurance and cost per square foot of rental space are normally distributed.

Ans: True

5. Normal distribution is a skewed distribution with its tails extending to infinity on either side of the mean.

Ans: False

6. Since a normal distribution curve extends from minus infinity to plus infinity, the area under the curve is infinity.

Ans: False

7. A z-score is the number of standard deviations that a value of a random variable is above or below the mean.

Ans: True

8. A normal distribution with a mean of zero and a standard deviation of 1 is called a null distribution.

Ans: False

9. A standard normal distribution has a mean of zero and a standard deviation of one.

Ans: True

10. The standard normal distribution is also called a finite distribution because its mean is zero and standard deviation one, always.

Ans: False

11. In a standard normal distribution, if the area under curve to the right of a z-value is 0.10, then the area to the left of the same z-value is -0.10.

Ans: False

12. Binomial distributions in which the sample sizes are large may be approximated by a Poisson distribution.

Ans: False

13. A correction for continuity must be made when approximating the binomial distribution problems using a normal distribution.

Ans: True

14. If arrivals at a bank followed a Poisson distribution, then the time between arrivals would follow a binomial distribution.

Ans: False

15. For an exponential distribution, the mean is always equal to its variance.

Ans: False

Multiple Choice

16. If *x* is uniformly distributed over the interval 8 to 12, inclusively (8 £ *x* £ 12), then the height of this distribution, f(*x*), is __________________.

a) 1/8

b) 1/4

c) 1/12

d) 1/20

e) 1/24

Ans: b

17. If *x* is uniformly distributed over the interval 8 to 12, inclusively (8 £ *x* £ 12), then the mean of this distribution is __________________.

a) 10

b) 20

c) 5

d) 0

e) unknown

Ans: a

18. If *x* is uniformly distributed over the interval 8 to 12, inclusively (8 £ *x* £ 12), then the standard deviation of this distribution is __________________.

a) 4.00

b) 1.33

c) 1.15

d) 2.00

e) 1.00

Ans: c

19. If *x* is uniformly distributed over the interval 8 to 12, inclusively (8 £ *x* £ 12), then the probability, P(9 £ *x* £ 11), is __________________.

a) 0.250

b) 0.500

c) 0.333

d) 0.750

e) 1.000

Ans: b

20. If *x* is uniformly distributed over the interval 8 to 12, inclusively (8 £ *x* £ 12), then the probability, P(10.0 £ *x* £ 11.5), is __________________.

a) 0.250

b) 0.333

c) 0.375

d) 0.500

e) 0.750

Ans: c

21. If *x* is uniformly distributed over the interval 8 to 12, inclusively (8 £ *x* £ 12), then the probability, P(13 £ *x* £ 15), is __________________.

a) 0.250

b) 0.500

c) 0.375

d) 0.000

e) 1.000

Ans: d

22. If *x* is uniformly distributed over the interval 8 to 12, inclusively (8 £ *x* £ 12), then P(*x* < 7) is __________________.

a) 0.500

b) 0.000

c) 0.375

d) 0.250

e) 1.000

Ans: b

23. If *x* is uniformly distributed over the interval 8 to 12, inclusively (8 £ *x* £ 12), then P(*x* £ 11) is __________________.

a) 0.750

b) 0.000

c) 0.333

d) 0.500

e) 1.000

Ans: a

24. If *x* is uniformly distributed over the interval 8 to 12, inclusively (8 £ *x* £ 12), then P(*x* ³ 10) is __________________.

a) 0.750

b) 0.000

c) 0.333

d) 0.500

e) 0.900

Ans: d

25. If a continuous random variable *x* is uniformly distributed over the interval 8 to 12, inclusively, then P(*x* = exactly 10) is __________________.

a) 0.750

b) 0.000

c) 0.333

d) 0.500

e) 0.900

Ans: b

26. If *x*, the time (in minutes) to complete an oil change job at certain auto service station, is uniformly distributed over the interval 20 to 30, inclusively (20 £ *x* £ 30), then the height of this distribution, f(*x*), is __________________.

a) 1/10

b) 1/20

c) 1/30

d) 12/50

e) 1/60

Ans: a

27. If *x*, the time (in minutes) to complete an oil change job at certain auto service station, is uniformly distributed over the interval 20 to 30, inclusively (20 £ *x* £ 30), then the mean of this distribution is __________________.

a) 50

b) 25

c) 10

d) 15

e) 5

Ans: b

28. If *x*, the time (in minutes) to complete an oil change job at certain auto service station, is uniformly distributed over the interval 20 to 30, inclusively (20 £ *x* £ 30), then the standard deviation of this distribution is __________________.

a) unknown

b) 8.33

c) 0.833

d) 2.89

e) 1.89

Ans: d

29. If *x*, the time (in minutes) to complete an change job at certain auto service station, is uniformly distributed over the interval 20 to 30, inclusively (20 £ *x* £ 30), then the probability that an oil change job is completed in 25 to 28 minutes, inclusively, i.e., P(25 £ *x* £ 28) is __________________.

a) 0.250

b) 0.500

c) 0.300

d) 0.750

e) 81.000

Ans: c

30. If *x*, the time (in minutes) to complete an change job at certain auto service station, is uniformly distributed over the interval 20 to 30, inclusively (20 £ *x* £ 30), then the probability that an oil change job is completed in 21.75 to 24.75 minutes, inclusively, i.e., P(21.75 £ *x* £ 24.25) is __________________.

a) 0.250

b) 0.333

c) 0.375

d) 0.000

e) 1.000

Ans: a

31. If *x*, the time (in minutes) to complete an change job at certain auto service station, is uniformly distributed over the interval 20 to 30, inclusively (20 £ *x* £ 30), then the probability that an oil change job is completed in 33 to 35 minutes, inclusively, i.e., P(33 £ *x* £ 35) is __________________.

a) 0.5080

b) 0.000

c) 0.375

d) 0.200

e) 1.000

Ans: b

32. If *x*, the time (in minutes) to complete an change job at certain auto service station, is uniformly distributed over the interval 20 to 30, inclusively (20 £ *x* £ 30), then the probability that an oil change job is completed in less than 17 minutes, i.e., P(*x* < 17) is __________________.

a) 0.500

b) 0.300

c) 0.000

d) 0.250

e) 1.000

Ans: c

33. If *x*, the time (in minutes) to complete an change job at certain auto service station, is uniformly distributed over the interval 20 to 30, inclusively (20 £ *x* £ 30), then the probability that an oil change job is completed in less than or equal to 22 minutes, i.e., P(*x* £ 22) is __________________.

a) 0.200

b) 0.300

c) 0.000

d) 0.250

e) 1.000

Ans: a

34. If *x*, the time (in minutes) to complete an change job at certain auto service station, is uniformly distributed over the interval 20 to 30, inclusively (20 £ *x* £ 30), then the probability that an oil change job will be completed 24 minutes or more, i.e., P(*x* ³ 24) is __________________.

a) 0.100

b) 0.000

c) 0.333

d) 0.600

e) 1.000

Ans: d

35. The normal distribution is an example of _______.

a) a discrete distribution

b) a continuous distribution

c) a bimodal distribution

d) an exponential distribution

e) a binomial distribution

Ans: b

36. The total area underneath any normal curve is equal to _______.

a) the mean

b) one

c) the variance

d) the coefficient of variation

e) the standard deviation

Ans: b

37. The area to the left of the mean in any normal distribution is equal to _______.

a) the mean

b) 1

c) the variance

d) 0.5

e) -0.5

Ans: d

38. A standard normal distribution has the following characteristics:

a) the mean and the variance are both equal to 1

b) the mean and the variance are both equal to 0

c) the mean is equal to the variance

d) the mean is equal to 0 and the variance is equal to 1

e) the mean is equal to the standard deviation

Ans: d

39. If *x* is a normal random variable with mean 80 and standard deviation 5, the *z-*score for *x *= 88 is ________.

a) 1.8

b) -1.8

c) 1.6

d) -1.6

e) 8.0

Ans: c

40. Suppose *x* is a normal random variable with mean 60 and standard deviation 2. A *z* score was calculated for a number, and the *z* score is 3.4. What is *x*?

a) 63.4

b) 56.6

c) 68.6

d) 53.2

e) 66.8

Ans: e

41. Suppose *x* is a normal random variable with mean 60 and standard deviation 2. A *z* score was calculated for a number, and the *z* score is -1.3. What is *x*?

a) 58.7

b) 61.3

c) 62.6

d) 57.4

e) 54.7

Ans: d

42. Let *z* be a normal random variable with mean 0 and standard deviation 1. What is P(*z* < 1.3)?

a) 0.4032

b) 0.9032

c) 0.0968

d) 0.3485

e) 0. 5485

Ans: b

43. Let *z* be a normal random variable with mean 0 and standard deviation 1. What is P(1.3 < *z* < 2.3)?

a) 0.4032

b) 0.9032

c) 0.4893

d) 0.0861

e) 0.0086

Ans: d

44. Let *z* be a normal random variable with mean 0 and standard deviation 1. What is P(*z* > 2.4)?

a) 0.4918

b) 0.9918

c) 0.0082

d) 0.4793

e) 0.0820

Ans: c

45. Let *z* be a normal random variable with mean 0 and standard deviation 1. What is P(*z* < -2.1)?

a) 0.4821

b) -0.4821

c) 0.9821

d) 0.0179

e) -0.0179

Ans: d

46. Let *z* be a normal random variable with mean 0 and standard deviation 1. What is P(*z* > -1.1)?

a) 0.36432

b) 0.8643

c) 0.1357

d) -0.1357

e) -0.8643

Ans: b

47. Let *z* be a normal random variable with mean 0 and standard deviation 1. What is

P(-2.25 < *z* < -1.1)?

a) 0.3643

b) 0.8643

c) 0.1235

d) 0.4878

e) 0.5000

Ans: c

48. Let *z* be a normal random variable with mean 0 and standard deviation 1. The 50^{th} percentile of *z* is ____________.

a) 0.6700

b) -1.254

c) 0.0000

d) 1.2800

e) 0.5000

Ans: c

49. Let *z* be a normal random variable with mean 0 and standard deviation 1. The 75^{th} percentile of *z* is ____________.

a) 0.6700

b) -1.254

c) 0.0000

d) 1.2800

e) 0.5000

Ans: a

50. Let *z* be a normal random variable with mean 0 and standard deviation 1. The 90^{th} percentile of *z* is ____________.

a) 1.645

b) -1.254

c) 1.960

d) 1.280

e) 1.650

Ans: d

51. A *z* score is the number of __________ that a value is from the mean.

a) variances

b) standard deviations

c) units

d) miles

e) minutes

Ans: b

52. Within a range of *z* scores from -1 to +1, you can expect to find _______ per cent of the values in a normal distribution.

a) 95

b) 99

c) 68

d) 34

e) 100

Ans: c

53. Within a range of *z* scores from -2 to +2, you can expect to find _______ per cent of the values in a normal distribution.

a) 95

b) 99

c) 68

d) 34

e) 100

Ans: a

54. The expected (mean) life of a particular type of light bulb is 1,000 hours with a standard deviation of 50 hours. The life of this bulb is normally distributed. What is the probability that a randomly selected bulb would last longer than 1150 hours?

a) 0.4987

b) 0.9987

c) 0.0013

d) 0.5013

e) 0.5513

Ans: c

55. The expected (mean) life of a particular type of light bulb is 1,000 hours with a standard deviation of 50 hours. The life of this bulb is normally distributed. What is the probability that a randomly selected bulb would last fewer than 1100 hours?

a) 0.4772

b) 0.9772

c) 0.0228

d) 0.5228

e) 0.5513

Ans: b

56. The expected (mean) life of a particular type of light bulb is 1,000 hours with a standard deviation of 50 hours. The life of this bulb is normally distributed. What is the probability that a randomly selected bulb would last fewer than 940 hours?

a) 0.3849

b) 0.8849

c) 0.1151

d) 0.6151

e) 0.6563

Ans: c

57. Suppose you are working with a data set that is normally distributed with a mean of 400 and a standard deviation of 20. Determine the value of *x* such that 60% of the values are greater than *x*.

a) 404.5

b) 395.5

c) 405.0

d) 395.0

e) 415.0

Ans: d

58. Sure Stone Tire Company has established that the useful life of a particular brand of its automobile tires is normally distributed with a mean of 40,000 miles and a standard deviation of 5000 miles. What is the probability that a randomly selected tire of this brand has a life of at most 30,000 miles?

a) 0.5000

b) 0.4772

c) 0.0228

d) 0.9772

e) 1.0000

Ans: c

59. Sure Stone Tire Company has established that the useful life of a particular brand of its automobile tires is normally distributed with a mean of 40,000 miles and a standard deviation of 5000 miles. What is the probability that a randomly selected tire of this brand has a life of at least 50,000 miles?

a) 0.0228

b) 0.9772

c) 0.5000

d) 0.4772

e) 1.0000

Ans: a

60. Sure Stone Tire Company has established that the useful life of a particular brand of its automobile tires is normally distributed with a mean of 40,000 miles and a standard deviation of 5000 miles. What is the probability that a randomly selected tire of this brand has a life between 30,000 and 50,000 miles?

a) 0.5000

b) 0.4772

c) 0.9544

d) 0.9772

e) 1.0000

Ans: c

61. The net profit from a certain investment is normally distributed with a mean of $10,000 and a standard deviation of $5,000. The probability that the investor will not have a net loss is _____________.

a) 0.4772

b) 0.0228

c) 0.9544

d) 0.9772

e) 1.0000

Ans: d

62. The net profit of an investment is normally distributed with a mean of $10,000 and a standard deviation of $5,000. The probability that the investor’s net gain will be at least $5,000 is _____________.

a) 0.1859

b) 0.3413

c) 0.8413

d) 0.4967

e) 0.5000

Ans: c

63. Completion time (from start to finish) of a building remodeling project is normally distributed with a mean of 200 work-days and a standard deviation of 10 work-days. The probability that the project will be completed within 185 work-days is ______.

a) 0.0668

b) 0.4332

c) 0.5000

d) 0.9332

e) 0.9950

Ans: a

64. Completion time (from start to finish) of a building remodeling project is normally distributed with a mean of 200 work-days and a standard deviation of 10 work-days. To be 99% sure that we will not be late in completing the project, we should request a completion time of _______ work-days.

a) 211

b) 207

c) 223

d) 200

e) 250

Ans: c

65. Let *x* be a binomial random variable with* n*=20 and* p*=.8. If we use the normal distribution to approximate probabilities for this, we would use a mean of _______.

a) 20

b) 16

c) 3.2

d) 8

e) 5

Ans: b

66. Let *x* be a binomial random variable with* n*=20 and* p*=.8. If we use the normal distribution to approximate probabilities for this, a correction for continuity should be made. To find the probability of more than 12 successes, we should find _______.

a) P(*x*>12.5)

b) P(*x*>12)

c) P(*x*>11.5)

d) P(*x*__<__11.5)

e) P(*x* < 12)

Ans: a

67. The exponential distribution is an example of _______.

a) a discrete distribution

b) a continuous distribution

c) a bimodal distribution

d) a normal distribution

e) a symmetrical distribution

Ans: b

68. For an exponential distribution with a lambda (l) equal to 4, the standard deviation equal to _______.

a) 4

b) 0.5

c) 0.25

d) 1

e) 16

Ans: c

69. The average time between phone calls arriving at a call center is 30 seconds. Assuming that the time between calls is exponentially distributed, find the probability that more than a minute elapses between calls.

a) 0.135

b) 0.368

c) 0.865

d) 0.607

e) 0.709

Ans: a

70. The average time between phone calls arriving at a call center is 30 seconds. Assuming that the time between calls is exponentially distributed, find the probability that less than two minutes elapse between calls.

a) 0.018

b) 0.064

c) 0.936

d) 0.982

e) 1.000

Ans: d

71. At a certain workstation in an assembly line, the time required to assemble a component is exponentially distributed with a mean time of 10 minutes. Find the probability that a component is assembled in 7 minutes or less?

a) 0.349

b) 0.591

c) 0.286

d) 0.714

e) 0.503

Ans: e

72. At a certain workstation in an assembly line, the time required to assemble a component is exponentially distributed with a mean time of 10 minutes. Find the probability that a component is assembled in 3 to 7 minutes?

a) 0.5034

b) 0.2592

c) 0.2442

d) 0.2942

e) 0.5084

Ans: c

73. On Saturdays, cars arrive at Sam Schmitt's Scrub and Shine Car Wash at the rate of 6 cars per fifteen minute interval. The probability that at least 2 minutes will elapse between car arrivals is _____________.

a) 0.0000

b) 0.4493

c) 0.1353

d) 1.0000

e) 1.0225

Ans: b

74. On Saturdays, cars arrive at Sam Schmitt's Scrub and Shine Car Wash at the rate of 6 cars per fifteen minute interval. The probability that less than 10 minutes will elapse between car arrivals is _____________.

a) 0.8465

b) 0.9817

c) 0.0183

d) 0.1535

e) 0.2125

Ans: b

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