Deck 7: Continuous Probability Distributions

Question 1

A continuous uniform distribution U(0, 800) will have μ = 400 and σ = 230.94.

Answer 1

True

Question 2

The mean, median, and mode of a normal distribution will always be the same.

Answer 2

True

Question 3

We would use a normal distribution to model the waiting time until the next Florida hurricane strike.

Answer 3

False

Question 4

For a continuous uniform distribution U(200, 400), the parameters are μ = 300 and σ = 100.

Question 5

The triangular distribution is symmetric.

Question 6

There is a simple formula for normal areas, but we prefer a table for greater accuracy.

Question 7

If arrivals follow a Poisson distribution, waiting times follow the exponential distribution.

Question 8

A continuous uniform distribution U(100, 200) will have the same standard deviation as a continuous uniform distribution U(200,300).

Question 9

A continuous uniform distribution is always symmetric.

Question 10

The exponential distribution is continuous and the Poisson distribution is discrete, yet the two distributions are closely related.

Question 11

A triangular distribution can be skewed either left or right.

Question 12

The height and width of a continuous uniform distribution's PDF are the same.

Question 13

The triangular distribution T(0, 10, 20) is skewed left.

Question 14

Any normal distribution has a mean of 0 and a standard deviation of 1.

Question 15

For a continuous random variable, the total area beneath the PDF will be greater than zero but less than one.

Question 16

Normal distributions differ only in their means and variances.

Question 17

The triangular distribution is used in "what-if" analysis for business planning.

Question 18

Experience suggests that 4 percent of all college students had a tonsillectomy. In a sample of 300 college students, we need to find the probability that at least 10 had a tonsillectomy. It is acceptable to use the normal distribution to estimate this probability.

Question 19

The exponential distribution describes the number of arrivals per unit of time.

Question 20

The exponential distribution is always skewed right.

Question 21

A random variable X is best described by a continuous uniform distribution from 20 to 45 inclusive. What is P(30 ≤ X ≤ 40)?

A).20
B).40
C).60
D).80

Question 22

A machine dispenses water into a glass. Assuming that the amount of water dispensed follows a continuous uniform distribution from 10 ounces to 16 ounces, the standard deviation of the amount of water dispensed is about:

A)1.73 ounces.
B)3.00 ounces.
C)0.57 ounce.
D)3.51 ounces.

Question 23

If arrivals occur at a mean rate of 3.6 events per hour, the exponential probability of waiting less than 0.5 hour for the next arrival is:

A).7122.
B).8105.
C).8347.
D).7809.

Question 24

If arrivals occur at a mean rate of 2.6 events per minute, the exponential probability of waiting more than 1.5 minutes for the next arrival is:

A).0202.
B).0122.
C).0535.
D).2564.

Question 25

The true proportion of accounts receivable with some kind of error is 4 percent for Venal Enterprises. If an auditor randomly samples 50 accounts receivable, it is acceptable to use the normal approximation to estimate the probability that fewer than two will contain errors.

Question 26

A random variable X is best described by a continuous uniform distribution from 20 to 45 inclusive. The mean of this distribution is:

A)30.5.
B)31.5.
C)32.5.
D)33.5.

Question 27

A random variable X is best described by a continuous uniform distribution from 20 to 45 inclusive. The standard deviation of this distribution is approximately:

A)52.1.
B)32.5.
C)6.85.
D)7.22.

Question 28

The number of lightning strikes in a day in Miami is a continuous random variable.

Question 29

If arrivals occur at a mean rate of 3.6 events per hour, the exponential probability of waiting more than 0.5 hour for the next arrival is:

A).2407.
B).1653.
C).1222.
D).5000.

Question 30

The area under an exponential curve can exceed 1 because the distribution is right-skewed.

Question 31

A machine dispenses water into a glass. Assuming that the amount of water dispensed follows a continuous uniform distribution from 10 ounces to 16 ounces, what is the probability that 13 or more ounces will be dispensed in a given glass?

A).1666
B).3333
C).5000
D).6666

Question 32

The normal distribution is a good approximation to the binomial if both π ≥ 10 and n ≥ 10.

Question 33

The normal distribution is a good approximation to the binomial if n = 200 and π = .03.

Question 34

If arrivals occur at a mean rate of 1.6 events per minute, the exponential probability of waiting less than 1 minute for the next arrival is:

A).2019.
B).7104.
C).8812.
D).7981.

Question 35

The Excel function =40*RAND() would generate random numbers with standard deviation approximately equal to

A)13.33.
B)20.00.
C)11.55.
D)19.27.

Question 36

The exponential distribution can be either right-skewed or left-skewed, depending on λ

Question 37

The area under a normal curve is 1 only if the distribution is standardized N(0, 1).

Question 38

A machine dispenses water into a glass. Assuming that the amount of water dispensed follows a continuous uniform distribution from 10 ounces to 16 ounces, the average amount of water dispensed by the machine is:

A)12 ounces.
B)13 ounces.
C)14 ounces.
D)16 ounces.

Question 39

The normal is a good approximation to the binomial when n is greater than or equal to 10.

Question 40

The normal distribution is a good approximation to the binomial if n = 25 and π = .50.

Question 41

The area under the normal curve between z = 0 and z = 1 is ______________ the area under the normal curve between z = 1 and z = 2.

A)less than
B)greater than
C)equal to

Question 42

Exam scores were normal in BIO 200. Jason's exam score was one standard deviation above the mean. What percentile is he in?

A)68^th
B)75^th
C)78^th
D)84^th

Question 43

A large number of applicants for admission to graduate study in business are given an aptitude test. Scores are normally distributed with a mean of 460 and standard deviation of 80. What fraction of applicants would you expect to have scores of 600 or above?

A)0.0401
B)0.4599
C)0.5401
D)0.0852

Question 44

In Melanie's Styling Salon, the time to complete a simple haircut is normally distributed with a mean of 25 minutes and a standard deviation of 4 minutes. What percentage of customers require less than 32 minutes for a simple haircut?

A)95.99 percent
B)99.45 percent
C)97.72 percent
D)45.99 percent Using Excel =NORMDIST(32,25,4,1) = 0.9599, or use z = (32 - 25)/4 = 1.75 with AppendixC.

Question 45

The lengths of brook trout caught in a certain Colorado stream are normally distributed with a mean of 14 inches and a standard deviation of 3 inches. What proportion of brook trout caught will be between 12 and 18 inches in length?

A).6563
B).6826
C).2486
D).4082

Question 46

The lengths of brook trout caught in a certain Colorado stream are normally distributed with a mean of 14 inches and a standard deviation of 3 inches. What lower limit should the State Game Commission set on length if it is desired that 80 percent of the catch may be kept by fishers?

A)12.80 inches
B)11.48 inches
C)12.00 inches
D)9.22 inches Using Excel =NORM.INV(.20,14,3) = 11.475, or X = 14 - 0.84(3) = 11.48 using AppendixC.

Question 47

A large number of applicants for admission to graduate study in business are given an aptitude test. Scores are normally distributed with a mean of 460 and standard deviation of 80. What fraction of the applicants would you expect to have a score of 400 or above?

A)0.2734
B)0.7734
C)0.7266
D)0.7500

Question 48

The price-earnings ratio for firms in a given industry follows the normal distribution. In this industry, a firm whose price-earnings ratio has a standardized value of z = 1.00 is approximately in the highest ______ percent of firms in the industry.

A)16 percent
B)34 percent
C)68 percent
D)75 percent

Question 49

The lengths of brook trout caught in a certain Colorado stream are normally distributed with a mean of 14 inches and a standard deviation of 3 inches. The first quartile for the lengths of brook trout would be:

A)16.01 inches.
B)11.00 inches.
C)11.98 inches.
D)10.65 inches.Using Excel =NORM.INV(.25,14,3) = 11.98, or Q₁ = 14 - 0.675(3) = 11.975 using AppendixC.

Question 50

Compared to the area between z = 1.00 and z = 1.25, the area between z = 2.00 and z = 2.25 in the standard normal distribution will be:

A)smaller.
B)larger.
C)the same.
D)impossible to compare without knowing μ and σ.

Question 51

Bob's z-score for the last exam was 1.52 in Prof. Axolotl's class BIO 417, "Life Cycle of the Ornithorhynchus." Bob said, "Oh, good, my score is in the top 10 percent." Assuming a normal distribution of scores, is Bob right?

A)Yes.
B)No.
C)Must have n to answer.

Question 52

The time required for a citizen to complete the 2010 U.S. Census "long" form is normally distributed with a mean of 40 minutes and a standard deviation of 10 minutes. What is the third quartile (in minutes) for the time required to complete the form?

A)44.75
B)46.75
C)47.50
D)52.50 Using Excel =NORM.INV(.75,40,10) = 46.75, or Q₃ = 40 + 0.675(10) = 46.75 using AppendixC.

Question 53

The probability is .80 that a standard normal random variable is between -z and +z. The value of z is approximately:

A)1.28.
B)1.35.
C)1.96.
D)1.45.

Question 54

The MPG (miles per gallon) for a certain compact car is normally distributed with a mean of 31 and a standard deviation of 0.8. What is the probability that the MPG for a randomly selected compact car would be less than 32?

A)0.3944
B)0.8944
C)0.1056
D)0.5596

Question 55

If the random variable Z has a standard normal distribution, then P(1.25 ≤ Z ≤ 2.17) is:

A)0.0906.
B)0.9200.
C)0.4700.
D)0.3944.

Question 56

The time required for a citizen to complete the 2010 U.S. Census "long" form is normally distributed with a mean of 40 minutes and a standard deviation of 10 minutes. The slowest 10 percent of the citizens would need at least how many minutes to complete the form?

A)27.2
B)35.8
C)52.8
D)59.6 Using Excel =NORM.INV(.90,40,10) = 52.82, or 40 + 1.282(10) = 52.82 using AppendixC.

Question 57

In Melanie's Styling Salon, the time to complete a simple haircut is normally distributed with a mean of 25 minutes and a standard deviation of 4 minutes. For a simple haircut, the middle 90 percent of the customers will require:

A)between 18.4 and 31.6 minutes.
B)between 19.9 and 30.1 minutes.
C)between 20.0 and 30.0 minutes.
D)between 17.2 and 32.8 minutes.

Question 58

In Melanie's Styling Salon, the time to complete a simple haircut is normally distributed with a mean of 25 minutes and a standard deviation of 4 minutes. The slowest quartile of customers will require longer than how many minutes for a simple haircut?

A)3(n + 1)/4 minutes
B)26 minutes
C)25.7 minutes
D)27.7 minutes Using Excel =NORM.INV(.75,25,4) = 27.698, or Q₃ = 25 + 0.675(4) = 27.7 using AppendixC.

Question 59

A student's grade on an examination was transformed to a z value of 0.67. Assuming a normal distribution, we know that she scored approximately in the top:

A)15 percent.
B)50 percent.
C)40 percent.
D)25 percent.

Question 60

The time required for a citizen to complete the 2010 U.S. Census "long" form is normally distributed with a mean of 40 minutes and a standard deviation of 10 minutes. What proportion of the citizens will require less than one hour?

A)0.4772
B)0.9772
C)0.9974
D)0.9997

Question 61

Bob's z-score for the last exam was -1.15 in FIN 417, "Capital Budgeting Strategies." Bob said, "Yipe! My score is within the bottom quartile." Assuming a normal distribution, is Bob right?

A)Yes
B)No
C)Must know the class size to answer

Question 62

What are the mean and standard deviation for the standard normal distribution?

A)μ = 0, σ = 0
B)μ = 1, σ = 1
C)μ = 1, σ = 0
D)μ = 0, σ = 1

Question 63

Which probability model is most appropriate to describe the waiting time (working days) until an office photocopier breaks down (i.e., requires unscheduled maintenance)?

A)Normal
B)Uniform
C)Exponential
D)Poisson

Question 64

Which model best describes your waiting time until you get the next nonworking web URL ("This page cannot be displayed") as you click on various websites for Florida condo rentals?

A)Triangular
B)Uniform
C)Normal
D)Exponential

Question 65

Any two normal curves are the same except for their:

A)standard deviations.
B)means.
C)standard deviations and means.
D)standard deviations, means, skewness, and kurtosis.

Question 66

Assume that X is normally distributed with a mean μ = $64. Given that P(X ≥ $75) = 0.2981, we can calculate that the standard deviation of X is approximately:

A)$20.76.
B)$13.17.
C)$5.83.
D)$7.05.

Question 67

The random variable X is normally distributed with mean of 80 and variance of 36. The 67^th percentile of the distribution is:

A)72.00.
B)95.84.
C)90.00.
D)82.64.

Question 68

If the mean time between unscheduled maintenance of LCD displays in a hospital's CT scan facility is 4,000 operating hours, what is the probability of unscheduled maintenance in the next 5,000 hours?

A).8000
B).7135
C).2865
D).5000

Question 69

A certain assembly line at Vexing Manufacturing Company averages 30 minutes between breakdowns. What is the probability that less than 6 minutes will elapse before the next breakdown?

A).8187
B).0488
C).1813
D).2224

Question 70

The area under the normal curve between the 20^th and 70^th percentiles is equal to:

A)0.7000.
B)0.5000.
C) 0.9193.
Logically, this must be .70 - .20 = .50, as you can verify from Appendix

Question 71

On average, 15 minutes elapse between discoveries of fraudulent corporate tax returns in a certain IRS office. What is the probability that less than 30 minutes will elapse before the next fraudulent corporate tax return is discovered?

A).1353
B).6044
C).7389
D).8647

Question 72

A certain assembly line at Vexing Manufacturing Company averages 30 minutes between breakdowns. The median time between breakdowns is:

A)30.0 minutes.
B)35.7 minutes.
C)25.4 minutes.
D)20.8 minutes.

Question 73

Light bulbs are normally distributed with an average lifetime of 1000 hours and a standard deviation of 250 hours. The probability that a light bulb picked at random will last less than 1500 hours is about:

A)97.72 percent.
B)95.44 percent.
C)75.00 percent.
D)68.00 percent.

Question 74

Regarding continuous probability distributions, which statement is incorrect?

A)The triangular distribution may be skewed left or right.
B)The uniform distribution is never skewed.
C)The normal distribution is sometimes skewed.
D)The exponential distribution is always skewed right.

Question 75

If the random variable Z has a standard normal distribution, then P(Z ≤ -1.37) is:

A)0.9147.
B)0.4147.
C)0.5016.
D)0.0853.

Question 76

If the mean time between in-flight aircraft engine shutdowns is 12,500 operating hours, the 90^th percentile of waiting times to the next shutdown will be:

A)20,180 hours.
B)28,782 hours.
C)23,733 hours.
D)18,724 hours.

Question 77

To convert a normally distributed variable X into a standard Z score we would:

A)subtract the mean from the original observation and divide the result by the variance.
B)subtract the mean from the original observation and divide the result by the standard deviation.
C)add the mean and the original observation, then divide by the variance.
D)subtract the mean from the standard deviation and divide by the variance.

Question 78

The variable in a normal distribution can assume any value between

A)-3 and +3.
B)-4 and +4.
C)-1 and +1.
D)-∞ and +∞.

Question 79

The standard deviation of a normal random variable X is $20. Given that P(X ≤ $10) = 0.1841. From this we can determine that the mean of the distribution is equal to:

A)$13.
B)$26.
C)$20.
D)$28.

Question 80

On average, a major earthquake (Richter scale 6.0 or above) occurs 3 times a decade in a certain California county. What is the probability that less than six months will pass before the next earthquake?

A).1393
B).8607
C).0952
D).9048