Deck 6: Continuous Probability Distributions

Question 1

The exponential distribution is related to the Poisson distribution.

Answer 1

True

Question 2

The probability density function of a continuous random variable is the counterpart to the probability mass function of a discrete random variable.

Answer 2

True

Question 3

The letter Z is used to denote a random variable with any normal distribution.

Answer 3

False

Question 4

Just as in the case of the continuous uniform distribution, the probability density function of the normal distribution may be easily used to compute probabilities.

Question 5

According the empirical rule for normally distributed variables, 75% of the values fall within one standard deviation of the mean.

Question 6

Examples of random variables that closely follow a normal distribution include the age and the class year designation of a college student.

Question 7

We are often interested in finding the probability that a continuous random variable assumes a particular value.

Question 8

Cumulative distribution functions can only be used to compute probabilities for continuous random variables.

Question 9

The mean of a continuous uniform distribution is simply the average of the upper and lower limits of the interval on which the distribution is defined.

Question 10

A continuous random variable is characterized by uncountable values and can take on any value within an interval.

Question 11

The continuous uniform distribution describes a random variable, defined on the interval
[a, b], that has an equally likely chance of assuming values within a specified range.

Question 12

Excel's function NORMAL.DIST can be used to compute probabilities for the normal distribution.

Question 13

The standard normal table is also referred to as the z table.

Question 14

A standard normal variable Z can be transformed to the normally distributed random variable X with only mean µ known.

Question 15

The mean and standard deviation of the continuous uniform distribution are equal.

Question 16

The normal probability distribution is symmetric and bell-shaped.

Question 17

Given that the probability distribution is normal, it is completely described by its mean μ > 0 and its standard deviation σ > 0.

Question 18

The probability density function for a continuous uniform distribution is positive for all values between -∞ and +∞.

Question 19

The standard normal distribution is a normal distribution with a mean equal to zero and a standard deviation equal to one.

Question 20

The lognormal distribution is clearly negatively skewed for σ > 1.

Question 21

The normal distribution is ________ in the sense that the tails get closer and closer to the horizontal axis but never touch it.

Question 22

Excel provides function ________ to compute probabilities for the exponential distribution.

Question 23

The exponential distribution is based entirely on one parameter called the ________ parameter.

Question 24

The waiting time at an elevator is uniformly distributed between 30 and 200 seconds. Find the mean and standard deviation of the waiting time.

A) 115 seconds and 49.07 seconds
B) 1.15 minutes and 0.4907 minutes
C) 1.15 minutes and 24.08333 (minute)^2
D) 115 seconds and 2408.3333 (second)²

Question 25

The time of a call to a technical support line is uniformly distributed between 2 and 10 minutes. What are the mean and variance of this distribution?

A) 6 minutes and 2.3094 (minutes)²
B) 6 minutes and 5.3333 (minutes)²
C) 6 minutes and 5.3333 minutes
D) 8 minutes and 2.3094 minutes

Question 26

The cumulative distribution function is denoted and defined as which of the following?

A) f(x) and f(x) = P(X ≤ x)
B) f(x) and f(x) = P(X ≥ x)
C) F(x) and F(x) = P(X ≤ x)
D) F(x) and F(x) = P(X ≥ x)

Question 27

A continuous random variable has the uniform distribution on the interval [a, b] if its probability density function f(x) ________.

A) Provides all probabilities for all x between a and b
B) Is bell-shaped between a and b
C) Is constant for all x between a and b, and 0 otherwise
D) Asymptotically approaches the x axis when x increases to +∞ or decreases to -∞

Question 28

The height of the probability density function f(x) of the uniform distribution defined on the interval [a, b] is ________.

A) 1/(b - a) between a and b, and zero otherwise
B) (b - a)/2 between a and b, and zero otherwise
C) (a + b)/2 between a and b, and zero otherwise
D) 1/(a + b) between a and b, and zero otherwise

Question 29

Which of the following does not represent a continuous random variable?

A) Height of oak trees in a park.
B) Heights and weights of newborn babies.
C) Time of a flight between Chicago and New York.
D) The number of customer arrivals to a bank between 10 am and 11 am.

Question 30

Which of the following is not a characteristic of a probability density function f(x)?

A) f(x) ≥ 0 for all values of x.
B) f(x) is symmetric around the mean.
C) The area under f(x) over all values of x equals one.
D) f(x) becomes zero or approaches zero if x increases to +infinity or decreases to -infinity.

Question 31

The waiting time at an elevator is uniformly distributed between 30 and 200 seconds. What is the probability a rider must wait between 1 minute and 1.5 minutes?

A) 0.1765
B) 0.3529
C) 0.5294
D) 0.8824

Question 32

We can use the ________ transformation, x = µ + zσ, to compute x values for given probabilities.

Question 33

The waiting time at an elevator is uniformly distributed between 30 and 200 seconds. What is the probability a rider waits less than two minutes?

A) 0.4706
B) 0.5294
C) 0.6000
D) 0.7059

Question 34

If the mean and the standard deviation of the underlying normal random variable equals respectively µ = 2 and σ = 1, the mean of a lognormal random variable equals ________.

Question 35

Which of the following is correct?

A) A continuous random variable has a probability density function but not a cumulative distribution function.
B) A discrete random variable has a probability mass function but not a cumulative distribution function.
C) A continuous random variable has a probability mass function, and a discrete random variable has a probability density function.
D) A continuous random variable has a probability density function, and a discrete random variable has a probability mass function.

Question 36

The waiting time at an elevator is uniformly distributed between 30 and 200 seconds. What is the probability a rider must wait more than 1.5 minutes?

A) 0.3529
B) 0.4500
C) 0.5294
D) 0.6471

Question 37

For σ < 1, the lognormal distribution somewhat resembles the ________ distribution.

Question 38

The z table provides ________ probabilities for positive and negative values of z.

Question 39

Scores on a business statistics final exam are normally distributed with a mean of 74 and standard deviation of 8. z value for the exam score of 84 equals ________.

Question 40

The cumulative distribution function F(x) of a continuous random variable X with the probability density function f(x) is which of the following?

A) The area under f over all values x.
B) The area under f over all values that are x or less.
C) The area under f over all values that are x or more.
D) The area under f over all non-negative values that are x or less.

Question 41

Find the probability P(−1.96 ≤ Z ≤ 1.96).

A) 0.0500
B) 0.9500
C) 0.9750
D) 1.9500

Question 42

The probability that a normal random variable is less than its mean is ________.

A) 0.0
B) 0.5
C) 1.0
D) Cannot be determined

Question 43

It is known that the length of a certain product X is normally distributed with μ = 20 inches and σ = 4 inches. How is the probability P(X > 28) related to P(X < 16)?

A) P(X > 28) is greater than (X < 16).
B) P(X > 28) is smaller than (X < 16).
C) P(X > 28) is the same as P(X < 16).
D) No comparison can be made with the given information.

Question 44

You work in marketing for a company that produces work boots. Quality control has sent you a memo detailing the length of time before the boots wear out under heavy use. They find that the boots wear out in an average of 208 days, but the exact amount of time varies, following a normal distribution with a standard deviation of 14 days. For an upcoming ad campaign, you need to know the percent of the pairs that last longer than six months-that is, 180 days. Use the empirical rule to approximate this percent.

A) 2.5%
B) 5%
C) 95%
D) 97.5%

Question 45

It is known that the length of a certain product X is normally distributed with μ = 20 inches. How is the probability P(X < 20) related to P(X < 16)?

A) P(X < 20) is greater than P(X < 16).
B) P(X < 20) is smaller than P(X < 16).
C) P(X < 20) is the same as P(X < 16).
D) No comparison can be made with the given information.

Question 46

What does it mean when we say that the tails of the normal curve are asymptotic to the x axis?

A) The tails get closer and closer to the x axis but never touch it.
B) The tails get closer and closer to the x axis and eventually touch it.
C) The tails get closer and closer to the x axis and eventually cross this axis.
D) The tails get closer and closer to the x axis and eventually become this axis.

Question 47

Find the probability P(−1.96 ≤ Z ≤ 0).

A) 0.0250
B) 0.0500
C) 0.4750
D) 0.5250

Question 48

Alex is in a hurry to get to work and is rushing to catch the bus. She knows that the bus arrives every six minutes during rush hour, but does not know the exact times the bus is due. She realizes that from the time she arrives at the stop, the amount of time that she will have to wait follows a uniform distribution with a lower bound of 0 minutes and an upper bound of six minutes. What is the probability that she will have to wait more than two minutes?

A) 0.1667
B) 0.3333
C) 0.6667
D) 1.0000

Question 49

Find the z value such that P(Z ≤ z) = 0.9082.

A) z = −1.33
B) z = 0.1814
C) z = 0.8186
D) z = 1.33

Question 50

An analyst is forecasting net income for Excellence Corporation for the next fiscal year. Her low-end estimate of net income is $250,000, and her high-end estimate is $350,000. Prior research allows her to assume that net income follows a continuous uniform distribution. The probability that net income will be greater than or equal to $337,500 is ________.

A) 12.5%
B) 29.6%
C) 87.5%
D) 96.4%

Question 51

Find the z value such that P(−z ≤ Z ≤ z) = 0.95.

A) z = −1.645
B) z = −1.96
C) z = 1.645
D) z = 1.96

Question 52

The probability P(Z < −1.28) is closest to ________.

A) −0.10
B) 0.10
C) 0.20
D) 0.90

Question 53

Let X be normally distributed with mean μ and standard deviation σ > 0. Which of the following is true about the z value corresponding to a given x value?

A) A positive z = (x - μ)/σ indicates how many standard deviations x is above μ.
B) A negative z = (x - μ)/σ indicates how many standard deviations x is below μ.
C) The z value corresponding to x = μ is zero.
D) All of the above.

Question 54

Suppose the average price of gasoline for a city in the United States follows a continuous uniform distribution with a lower bound of $3.50 per gallon and an upper bound of $3.80 per gallon. What is the probability a randomly chosen gas station charges less than $3.70 per gallon?

A) 0.3000
B) 0.3333
C) 0.6667
D) 1.0000

Question 55

Suppose the average price of gasoline for a city in the United States follows a continuous uniform distribution with a lower bound of $3.50 per gallon and an upper bound of $3.80 per gallon. What is the probability a randomly chosen gas station charges between $3.70 and $3.90 per gallon?

A) 0.3000
B) 0.3333
C) 0.6667
D) 1.0000

Question 56

It is known that the length of a certain product X is normally distributed with μ = 20 inches. How is the probability P(X > 24) related to P(X < 16)?

A) P(X > 24) is greater than P(X < 16).
B) P(X > 24) is smaller than P(X < 16).
C) P(X > 24) is the same as P(X < 16).
D) No comparison can be made with the given information.

Question 57

The probability P(Z > 1.28) is closest to ________.

A) −0.10
B) 0.10
C) 0.20
D) 0.90

Question 58

How many parameters are needed to fully describe any normal distribution?

A) 1
B) 2
C) 3
D) 4

Question 59

It is known that the length of a certain product X is normally distributed with μ = 20 inches. How is the probability P(X > 16) related to P(X < 16)?

A) P(X > 16) is greater than P(X < 16).
B) P(X > 16) is smaller than P(X < 16).
C) P(X > 16) is the same as P(X < 16).
D) No comparison can be made with the given information.

Question 60

Suppose the average price of gasoline for a city in the United States follows a continuous uniform distribution with a lower bound of $3.50 per gallon and an upper bound of $3.80 per gallon. What is the probability a randomly chosen gas station charges more than $3.70 per gallon?

A) 0.3000
B) 0.3333
C) 0.6667
D) 1.0000

Question 61

For any normally distributed random variable with mean μ and standard deviation σ, the proportion of the observations that fall outside the interval [μ - σ, μ + σ] is the closest to ________.

A) 0.0466
B) 0.3174
C) 0.8413
D) 0.1687

Question 62

Sarah's portfolio has an expected annual return at 8%, with an annual standard deviation at 12%. If her investment returns are normally distributed, then in any given year Sarah has an approximate ________.

A) 50% chance that the actual return will be greater than 8%
B) 68% chance that the actual return will fall within 4% and 20%
C) 68% chance that the actual return will fall within -20% and 20%
D) 95% chance that the actual return will fall within -4% and 28%

Question 63

A hedge fund returns on average 26% per year with a standard deviation of 12%. Using the empirical rule, approximate the probability the fund returns over 50% next year.

A) 0.5%
B) 1%
C) 2.5%
D) 5%

Question 64

Let X be normally distributed with mean µ = 25 and standard deviation σ = 5. Find the value x such that P(X ≥ x) = 0.1736.

A) −0.94
B) 0.94
C) 20.30
D) 29.70

Question 65

The stock price of a particular asset has a mean and standard deviation of $58.50 and $8.25, respectively. Use the normal distribution to compute the 95th percentile of this stock price.

A) −1.645
B) 1.645
C) 44.93
D) 72.07

Question 66

Sarah's portfolio has an expected annual return at 10%, with an annual standard deviation at 12%. If her investment returns are normally distributed, then in any given year Sarah has an approximate ________.

A) 50% chance that the actual return will be greater than 12%
B) 68% chance that the actual return will fall within -2% and 22%
C) 68% chance that the actual return will fall within -20% and 22%
D) 95% chance that the actual return will fall within -2% and 24%

Question 67

The time to complete the construction of a soapbox derby car is normally distributed with a mean of three hours and a standard deviation of one hour. Find the probability that it would take exactly 3.7 hours to construct a soapbox derby car.

A) 0.0000
B) 0.5000
C) 0.7580
D) 0.2420

Question 68

If X has a normal distribution with µ = 100 and σ = 5, then the probability P(90 ≤ X ≤ 95) can be expressed in terms of a standard normal variable Z as ________.

A) P(2 ≤ Z ≤ 1)
B) P(-2 ≤ Z ≤ 1)
C) P(-2 ≤ Z ≤ -1)
D) P(-2 ≤ Z ≤ -2)

Question 69

Gold miners in Alaska have found, on average, 12 ounces of gold per 1,000 tons of dirt excavated with a standard deviation of 3 ounces. Assume the amount of gold found per 1,000 tons of dirt is normally distributed. What is the probability the miners find more than 16 ounces of gold in the next 1,000 tons of dirt excavated?

A) 0.0912
B) 0.4082
C) 0.5918
D) 0.9082

Question 70

Let X be normally distributed with mean µ = 250 and standard deviation σ = 80. Find the value x such that P(X ≤ x) = 0.0606.

A) −1.55
B) 1.55
C) 126
D) 374

Question 71

The time to complete the construction of a soapbox derby car is normally distributed with a mean of three hours and a standard deviation of one hour. Find the probability that it would take between 2.5 and 3.5 hours to construct a soapbox derby car.

A) 0.3085
B) 0.3830
C) 0.6170
D) 0.6915

Question 72

Let X be normally distributed with mean µ = 250 and standard deviation σ = 80. Find the value x such that P(X ≤ x) = 0.9394.

A) −1.55
B) 1.55
C) 126
D) 374

Question 73

For any normally distributed random variable with mean μ and standard deviation σ, the percent of the observations that fall between [μ - 2σ, μ + 2σ] is the closest to ________.

A) 68%
B) 68.26%
C) 95%
D) 99.73%

Question 74

The starting salary of an administrative assistant is normally distributed with a mean of $50,000 and a standard deviation of $2,500. We know that the probability of a randomly selected administrative assistant making a salary between μ − x and μ + x is 0.7416. Find the salary range referred to in this statement.

A) $42,825 to $52,825
B) $42,825 to $57,175
C) $47,175 to $52,825
D) $47,175 to $57,175

Question 75

You are planning a May camping trip to Denali National Park in Alaska and want to make sure your sleeping bag is warm enough. The average low temperature in the park for May follows a normal distribution with a mean of 32°F and a standard deviation of 8°F. Above what temperature must the sleeping bag be suited such that the temperature will be too cold only 5% of the time?

A) −1.645
B) 1.645
C) 18.84
D) 45.16

Question 76

An investment consultant tells her client that the probability of making a positive return with her suggested portfolio is 0.90. What is the risk, measured by standard deviation that this investment manager has assumed in her calculation if it is known that returns from her suggested portfolio are normally distributed with a mean of 6%?

A) 1.28%
B) 4.69%
C) 6.00%
D) 10.0%

Question 77

You are planning a May camping trip to Denali National Park in Alaska and want to make sure your sleeping bag is warm enough. The average low temperature in the park for May follows a normal distribution with a mean of 32°F and a standard deviation of 8°F. One sleeping bag you are considering advertises that it is good for temperatures down to 25°F. What is the probability that this bag will be warm enough on a randomly selected May night at the park?

A) 0.1894
B) 0.3106
C) 0.8092
D) 0.8800

Question 78

You are planning a May camping trip to Denali National Park in Alaska and want to make sure your sleeping bag is warm enough. The average low temperature in the park for May follows a normal distribution with a mean of 32°F and a standard deviation of 8°F. An inexpensive bag you are considering advertises to be good for temperatures down to 38°F. What is the probability that the bag will not be warm enough?

A) 0.2266
B) 0.2734
C) 0.7500
D) 0.7734

Question 79

The time to complete the construction of a soapbox derby car is normally distributed with a mean of three hours and a standard deviation of one hour. Find the probability that it would take more than five hours to construct a soapbox derby car.

A) 0
B) 0.0228
C) 0.4772
D) 0.9772

Question 80

The salary of teachers in a particular school district is normally distributed with a mean of $50,000 and a standard deviation of $2,500. Due to budget limitations, it has been decided that the teachers who are in the top 2.5% of the salaries would not get a raise. What is the salary level that divides the teachers into one group that gets a raise and one that doesn't?

A) −1.96
B) 1.96
C) 45,100
D) 54,900