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Ch06 - Ch. 6 prep questions
Analytical Methods for Business (BNAD 277)
University of Arizona
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A continuous random variable is characterized by uncountable values and can take on any value within an interval. True False
We are often interested in finding the probability that a continuous random variable assumes a particular value. True False
The probability density function of a continuous random variable can be regarded as a counterpart of the probability mass function of a discrete random variable. True False
Cumulative distribution functions can only be used to compute probabilities for continuous random variables. True False
The continuous uniform distribution describes a random variable, defined on the interval [a, b], that has an equally likely chance of assuming values within any subinterval of [a, b] with the same length. True False
The probability density function of a continuous uniform distribution is positive for all values between -∞
and +∞. True False
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. True False
The mean and standard deviation of the continuous uniform distribution are equal. True False
The probability density function of a normal distribution is in general characterized by being symmetric and bell-shaped. True False
Examples of random variables that closely follow a normal distribution include the age and the class year designation of a college student. True False
Given that the probability distribution is normal, it is completely described by its mean μ > 0 and its
standard deviation σ > 0. True False
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. True False
The standard normal distribution is a normal distribution with a mean equal to zero and a standard deviation equal to one. True False
The letter Z is used to denote a random variable with any normal distribution. True False
The standard normal table is also referred to as the z table. True False
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 continuous random variable has a probability mass function, and a discrete random variable has a probability density function. D continuous random variable has a probability density function, and a discrete random variable has a probability mass function.
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.
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.
- 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)
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
A continuous random variable has the uniform distribution on the interval [a, b] if its probability density function f(x). A. Is symmetric around its mean 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 -∞
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
Suppose the average price of gasoline for a city in the United States follows a continuous uniform distribution with a lower bound of $3 per gallon and an upper bound of $3 per gallon. What is the probability a randomly chosen gas station charges more than $3 per gallon? A. 0. B. 0. C. 0. D. 1.
How many parameters are needed to fully describe any normal distribution? A. 1 B. 2 C. 3 D. 4
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 gets 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.
The probability that a normal random variable is less than its mean is. A. 0. B. 0. C. 1. D. Cannot be determined
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.
It is known that the length of a certain product X is normally distributed with μ = 20 inches. How is the
probability related to? A. is greater than. B. is smaller than. C. is the same as. D. No comparison can be made with the given information.
- It is known that the length of a certain product X is normally distributed with μ = 20 inches. How is the
probability related to? A. is greater than. B. is smaller than. C. is the same as. D. No comparison can be made with the given information.
- It is known that the length of a certain product X is normally distributed with μ = 20 inches. How is the
probability related to? A. is greater than. B. is smaller than. C. is the same as. D. No comparison can be made with the given information.
- 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 related to? A. is greater than. B. is smaller than. C. is the same as. D. No comparison can be made with the given information.
The probability P(Z < -1) is closest to. A. -0. B. 0. C. 0. D. 0.
The probability P(Z > 1) is closest to. A. -0. B. 0. C. 0. D. 0.
Find the probability P(-1 ≤ Z ≤ 0).
A. 0. B. 0. C. 0. D. 0.
Find the probability P(-1 ≤ Z ≤ 1). A. 0. B. 0. C. 0. D. 1.
Find the z value such that. A. z = -1. B. z = 0. C. z = 0. D. z = 1.
Find the z value such that. A. z = -1. B. z = -1. C. z = 1. D. z = 1.
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% B. 5% C. 95% D. 97%
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 hours to construct a soapbox derby car. A. 0. B. 0. C. 0. D. 0.
Let X be normally distributed with mean μ = 250 and standard deviation σ = 80. Find the value x -such
that P(X ≤ x) = 0. A. -1. B. 1. C. 126 D. 374
- Let X be normally distributed with mean μ = 250 and standard deviation σ = 80. Find the value x such
that P(X ≤ x) = 0. A. -1. B. 1. C. 126 D. 374
- Let X be normally distributed with mean μ = 25 and standard deviation σ = 5. Find the value x such that
P(X ≥ x) = 0. A. -0. B. 0. C. 20. D. 29.
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% 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. B. 1. C. 45, D. 54,
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. Find the salary range referred to in this statement. A. $42,825 to $52, B. $42,825 to $57, C. $47,175 to $52, D. $47,175 to $57,
An investment consultant tells her client that the probability of making a positive return with her suggested portfolio is 0. What is the risk, measured by standard deviation, that this investment manager has assumed in his calculation if it is known that returns from her suggested portfolio are normally distributed with a mean of 6%? A. 1% B. 4% C. 6% D. 10%
The stock price of a particular asset has a mean and standard deviation of $58 and $8, respectively. Use the normal distribution to compute the 95th percentile of this stock price. A. -1. B. 1. C. 44. D. 72.
Exhibit 6-1. 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.
Refer to Exhibit 6-1. 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. B. 0. C. 0. D. 0.
- Exhibit 6-1. 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.
Refer to Exhibit 6-1. 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. B. 0. C. 0. D. 0.
- Exhibit 6-1. 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.
Refer to Exhibit 6-1. Above what temperature must the sleeping bag be suited such that the temperature will be too cold only 5% of the time? A. -1. B. 1. C. 18. D. 45.
- Exhibit 6-2. Gold miners in Alaska have found, on average, 12 ounces of gold per 1000 tons of dirt excavated with a standard deviation of 3 ounces. Assume the amount of gold found per 1000 tons of dirt is normally distributed.
Refer to Exhibit 6-2. What is the probability the miners find more than 16 ounces of gold in the next 1000 tons of dirt excavated? A. 0. B. 0. C. 0. D. 0.
Let the time between two consecutive arrivals at a grocery store check-out line be exponentially distributed with a mean of three minutes. Find the probability that the next arrival does not occur until at least four minutes have passed since the last arrival. A. 0. B. 0. C. 0. D. 0.
Exhibit 6-3. Patients scheduled to see their primary care physician at a particular hospital wait, on average, an additional eight minutes after their appointment is scheduled to start. Assume the time that patients wait is exponentially distributed.
Refer to Exhibit 6-3. What is the probability a randomly selected patient will have to wait more than 10 minutes? A. 0. B. 0. C. 0. D. 0.
- Exhibit 6-3. Patients scheduled to see their primary care physician at a particular hospital wait, on average, an additional eight minutes after their appointment is scheduled to start. Assume the time that patients wait is exponentially distributed.
Refer to Exhibit 6-3. What is the probability a randomly selected patient will see the doctor within five minutes of the scheduled time? A. 0. B. 0. C. 0. D. 0.
- Exhibit 6-4. The average time between trades for a high-frequency trading investment firm is 40 seconds. Assume the time between trades is exponentially distributed.
Refer to Exhibit 6-4. What is the probability that the time between trades for a randomly selected trade and the one proceeding it is less than 20 seconds? A. 0. B. 0. C. 0. D. 0.
- Exhibit 6-4. The average time between trades for a high-frequency trading investment firm is 40 seconds. Assume the time between trades is exponentially distributed.
Refer to Exhibit 6-4. What is the probability that the time between trades for a randomly selected trade and the one proceeding it is more than a minute? A. 0. B. 0. C. 0. D. 0.
If has a lognormal distribution, what can be said of the distribution of the random variable X? A. X follows a normal distribution. B. X follows an exponential distribution. C. X follows a standard normal distribution. D. X follows a continuous uniform distribution.
Find the mean of the lognormal variable if the mean and standard deviation of the underlying normal variable are 2 and 0, respectively. A. 0. B. 2. C. 10. D. 11.
Find the variance of the lognormal variable if the mean and variance of the underlying normal variable are 2 and 1, respectively. A. 0 B. 12. C. 15. D. 255.
Exhibit 6-5. The mean travel time to work is 25 minutes (U. Census 2010). Further, suppose that commute time follows a log-normal distribution with a standard deviation of 10 minutes.
Refer to Exhibit 6-5. What is the probability a randomly selected U. worker has a commute time of more than half an hour? A. 25% B. 31% C. 68% D. 74%
- Exhibit 6-5. The mean travel time to work is 25 minutes (U. Census 2010). Further, suppose that commute time follows a log-normal distribution with a standard deviation of 10 minutes.
Refer to Exhibit 6-5. What is the probability a randomly selected U. worker has a commute time of less than 20 minutes? A. 30% B. 34% C. 65% D. 69%
- Exhibit 6-6. Let the lifetime of a new Jet Ski be represented by a lognormal variable, where X is normally distributed. Let the mean of the lifetime of the Jet Ski be six years with a standard deviation of three years.
Refer to Exhibit 6-6. What proportion of the Jet Skis will last less than seven years? A. 0. B. 0. C. 0. D. 0.
- Exhibit 6-6. Let the lifetime of a new Jet Ski be represented by a lognormal variable, where X is normally distributed. Let the mean of the lifetime of the Jet Ski be six years with a standard deviation of three years.
Refer to Exhibit 6-6. What proportion of the Jet Skis will last nine or more years? A. 0. B. 0. C. 0. D. 0.
- Suppose Jennifer is waiting for a taxi cab. A taxi cab's arrival time is equally likely at any constant time range in the next 12 minutes.
a. Calculate the expected arrival time. b. What is the probability that a taxi arrives in three minutes or less?
- Find the following probabilities for a standard normal random variable Z.
a. b. c. d.
- Find the following probabilities for a standard normal random variable Z.
a. b. c.
d.
- Find the value of z for which the standard normal random variable Z satisfies the following:
a.
b.
c. d.
- Given normally distributed random variable X with a mean of 10 and a variance of 4, find the following probabilities.
a. b. c. d.
- Given normally distributed random variable X with a mean of 12 and a standard deviation of 3, find the following probabilities.
a. b. c. d.
- A normal random variable X has a mean of 17 and a variance of 5.
a. Find the value x for which P(X ≤ x) = 0. b. Find the value of x for which P(X > x) = 0.
- A soft drink company fills two-liter bottles on several different lines of production equipment. The fill volumes are normally distributed with a mean of 1 liters and a variance of 0 (liter) 2.
a. Find the probability that a randomly selected two-liter bottle would contain between 1 and 2. liters. b. If X is the fill volume of a randomly selected two-liter bottle, find the value of x for which P(X > x) = 0.
100 weight of competition pumpkins at the Circleville Pumpkin Show in Circleville, Ohio, can be represented by a normal distribution with a mean of 703 pounds and a standard deviation of 347 pounds.
a. Find the probability that a randomly selected pumpkin weighs at least 1622 pounds. b. Find the probability that a randomly selected pumpkin weighs between 465 and 1622 pounds.
101 average annual percentage rate (APR) for credit cards held by U. consumers is approximately 15 percent ("Ouch - Credit Card APR Now Tops 15 Percent," Time, January 3, 2012). Suppose the APR for new credit card offers is normally distributed with a mean of 15% and a standard deviation of 4%. What APR must a credit card charge to be in the bottom 10% of all cards?
102 average annual inflation rate in the United States over the past 98 years is 3% and has a standard deviation of approximately 5% (Inflationdata). In 1980, the inflation rate was above 13%. If the annual inflation rate is normally distributed, what is the probability that inflation will be above 13% next year?
103 Japan Sumo Association has begun to measure the body fat of wrestlers to try to combat the growing problem of excessive obesity within the sport. As of 2010, the average wrestler weighed 412 pounds. Suppose the weights of sumo wrestlers are normally distributed, with a standard deviation of 37 pounds. What is the probability that a randomly selected wrestler weighs between 350 and 450 pounds?
104 producer has a history of making bad movies. The movies he has produced have averaged $160, dollars at the box office with a standard deviation of $185,000. He thinks his latest movie will be a huge hit. How much will the movie earn at the box office if it is only expected to earn that much or more 0% of the time? (Assume that the profit at the box office is normally distributed.)
105 the amount of time customers must wait to check bags at the ticketing counter in Boston Logan Airport is exponentially distributed with a mean of 14 minutes. What is the probability that a randomly selected customer will have to wait more than 20 minutes?
106 average wait time to see a doctor at a maternity ward is 16 minutes. What is the probability that a patient will have to wait between 20 and 30 minutes before seeing a doctor? Suppose the wait time is exponentially distributed.
107 average wait time at a McDonald's drive-through window is about three minutes ("The Doctor Will See You Eventually," The Wall Street Journal, October 18, 2010). Suppose the wait time is exponentially distributed. What is the probability that a randomly selected customer will have to wait no more than five minutes?
108 is waiting for a taxi cab. The average wait time for a taxi is six minutes. Suppose the wait time is exponentially distributed. What is the probability that a taxi arrives in three minutes or less?
114 mean household income of France is approximately 20,000 euros. Suppose the household income in France has a standard deviation of 10,000 euros and follows a log-normal distribution. Estimate the proportion of French households that have an income of more than 25,000 euros.
115 house prices in a rich community in Chicago be represented by , where X is normally distributed. Suppose the mean house price is $1 million and the standard deviation is $0 million. What is the proportion of the houses that are worth more than $2 million?
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ch06 Key
Ch06 - Ch. 6 prep questions
Course: Analytical Methods for Business (BNAD 277)
University: University of Arizona
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