Statistics Unit 2
Consider the following cumulative distribution function for the discrete random variable X. What is the probability that X equals 2?
.14 The probability distribution of a discrete random variable is a list of the values with the associated probabilities. The cumulative distribution function of X is defined as P(X ≤ x). The cumulative probability representation is convenient when we are interested in finding the probability over a range of values rather than a specific value. P(X = 2) = P(X ≤ 2) − P(X ≤ 1) = 0.44 − 0.30 = 0.14
What is the probability that X is greater than 3? x12345P(X ≤ x)0.100.350.750.851.00
.25
How many parameters are needed to fully describe any normal distribution? Multiple Choice 1 2 3 4
2
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 ______. Multiple Choice 68% 68.26% 95% 99.73%
95%
Which of the following represents a subjective probability? Multiple Choice The probability of rolling a 2 on a single die is one in six. Based on a conducted experiment, the probability of tossing a head on an unfair coin is 0.6. A skier believes she has a 10% chance of winning a gold medal. Based on past observation, a manager believes there is a three-in-five chance of retaining an employee for at least one year.
A skier believes she has a 10% chance of winning a gold medal. For well-defined problems an a priori probability can be calculated by reasoning about the problem. A subjective probability is based on personal experience and judgment.
What is a simple event? An event that contains all outcomes of a sample space An event that contains several outcomes of a sample space An event that contains only one outcome of a sample space An event that contains the most probable outcome of a sample space
An event that contains only one outcome of a sample space
Which of the following statements is the most accurate about a binomial random variable? Multiple Choice It has a bell-shaped distribution. It is a continuous random variable. It counts the number of successes in a given number of trials. It counts the number of successes in a specified time interval or region.
It counts the number of successes in a given number of trials. A binomial random variable is defined as the number of successes achieved in the n trials of a Bernoulli process.
Which of the following sets of outcomes described below in I and II represent mutually exclusive events? "Your final course grade is an A"; "Your final course grade is a B." "Your final course grade is an A"; "Your final course grade is a Pass."
Only I represents mutually exclusive events.
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)? Multiple Choice P(X < 20) is greater than P(X < 16). P(X < 20) is smaller than P(X < 16). P(X < 20) is the same as P(X < 16). No comparison can be made with the given information.
P(X < 20) is greater than P(X < 16).
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)? Multiple Choice P(X > 16) is greater than P(X < 16). P(X > 16) is smaller than P(X < 16). P(X > 16) is the same as P(X < 16). No comparison can be made with the given information.
P(X > 16) is greater than P(X < 16)
The cumulative distribution function F(x) of a continuous random variable X with the probability density function f(x) is which of the following? Multiple Choice The area under f over all values x. The area under f over all values that are x or less. The area under f over all values that are x or more. The area under f over all non-negative values that are x or less.
The area under f over all values that are x or less.
Which of the following is an example of a uniformly distributed random variable? Multiple Choice The waiting time at a checkout counter of a supermarket. The failure rate of a light bulb. The scheduled arrival time of a cable technician. The scores on the Business Statistics exam.
The scheduled arrival time of a cable technician.
A sample space contains outcomes of the relevant events. several outcomes of an experiment. all possible outcomes of an experiment. one of several outcomes of an experiment.
all possible outcomes of an experiment.
A probability based on logical analysis rather than on observation or personal judgment is best referred to as a(n) classical probability empirical probability subjective probability finite probability
classical probability
Mutually exclusive events contain all possible outcomes may share common outcomes do not share common outcomes do not contain all possible outcomes
do not share common outcomes
Which of the following is not a characteristic of a probability density function f(x)? Multiple Choice f(x) ≥ 0 for all values of x. f(x) is symmetric around the mean. The area under f(x) over all values of x equals one. f(x) becomes zero or approaches zero if x increases to +infinity or decreases to −infinity.
f(x) is symmetric around the mean.
A continuous random variable has the uniform distribution on the interval [a, b] if its probability density function f(x) ______. Multiple Choice provides all probabilities for all x between a and b is bell-shaped between a and b is constant for all x between a and b, and 0 otherwise asymptotically approaches the x axis when x increases to +∞ or decreases to −∞
is constant for all x between a and b, and 0 otherwise
We can think of the expected value of a random variable X as ________. Multiple Choice the long-run average of the random variable values generated over 100 independent repetitions the long-run average of the random variable values generated over 1,000 independent repetitions the long-run average of the random variable values generated over infinitely many independent repetitions the long-run average of the random variable values generated over a finite number of independent repetitions
the long-run average of the random variable values generated over infinitely many independent repetitions
What is the probability that a randomly selected customer prefers vanilla
.4
The probability that a normal random variable is less than its mean is ______. Multiple Choice 0.0 0.5 1.0 Cannot be determined
.5
The probability that a respondent is at least 30 years old is the closest to ______.
.50
5! is equal to 5 × 4 × 3 × 2 5 × 4 5 × 4 × 3 5 × 4 × 3 × 2 × 1
5 × 4 × 3 × 2 × 1
If A and B are independent events, which of the following is correct? Multiple Choice P(A∪B)=0P(A∪B)=0 P(A∩B)=0P(A∩B)=0 P(AB)=P(A)P(AB)=P(A) P(A∪B)=P(A)+P(B)
P(AB)=P(A) For two independent events A and B: P(A|B)=P(A)P(A|B)=P(A) .P(A∩B) = 0 and P(A∪B) = P(A) + P(B) both apply to mutually exclusive events.
What is the standard deviation of the number of homes sold by the realtor during a month? Multiple Choice 0.56 0.75 1 1.2
.75 The standard deviation of the discrete random variable X is calculated as SD(X)=σ=σ2−−√.SD(X)=σ=σ2.The variance of the discrete random variable X is calculated as Var(X) = σ2 = ∑(xi − μ)2 P(X = xi).E(X) = 0 × 0.20 + 1 × 0.40 + 2 × 0.40 = 1.2Var(X) = (0 − 1.2)2 × 0.20 + (1 − 1.2)2 × 0.40 + (2 − 1.2)2 × 0.40 = 0.56SD(X)=0.56−−−−√=0.75
The contingency table below provides frequencies for the preferred type of exercise for people under the age of 35 and those 35 years of age or older. Find the probability that an individual prefers running.
0.3915
What are the two key properties of a discrete probability distribution? Multiple Choice 0≤P (X = x) ≤1 and ∑P (X = xi ) = 0 0≤P (X = x) ≤1 and ∑P (X = xi ) = 1 −1≤P (X = x) ≤1 and ∑P (X = xi ) = 1 −1≤P (X = x) ≤1 and ∑P (X = xi ) = 0
0≤P (X = x) ≤1 and ∑P (X = xi ) = 1 The two key properties of discrete probability distributions are 0≤P (X = x) ≤1 and ∑P (X = xi ) = 1
The probability that a respondent read a book in the last month and is at least 30 years old is the closest to _______.
.33
What is the expected number of homes sold by the realtor during a month? Multiple Choice 1 1.2 1.5 2
1.2 The expected value of X is calculated as E(X) = μ = ∑xi P(X = xi)E(X) = 0 × 0.20 + 1 × 0.40 + 2 × 0.40 = 1.2
A daily mail is delivered to your house between 1:00 p.m. and 5:00 p.m. Assume delivery times follow the continuous uniform distribution. Determine the percentage of mail deliveries that are made after 4:00 p.m. Multiple Choice 25% 33.3% 37.5% 27.5%
25%
According to a study by the Centers for Disease Control and Prevention, about 33% of U.S. births are Caesarean deliveries. Suppose seven expectant mothers are randomly selected. The expected number of mothers who will not have a Caesarean delivery is ______. Multiple Choice 1.24 2.31 3.50 4.69
4.69 The expected value of a binomial random variable is calculated as E(X) = μ = np.μ = 7 × (1 − 0.33) = 4.69
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 less than two minutes? Multiple Choice 0.1667 0.3333 0.6667 1.0000
0.3333
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 ______. Multiple Choice 50% chance that the actual return will be greater than 8% 68% chance that the actual return will fall within 4% and 20% 68% chance that the actual return will fall within −20% and 20% 95% chance that the actual return will fall within −4% and 28%
50% chance that the actual return will be greater than 8%
What is probability? Any value between 0 and 1 is always treated as a probability of an event. A numerical value assigned to an event that measures the number of its occurrences. A value between 0 and 1 assigned to an event that measures the likelihood of its occurrence. A value between 0 and 1 assigned to an event that measures the unlikelihood of its occurrence.
A value between 0 and 1 assigned to an event that measures the likelihood of its occurrence.
The likelihood of Company A's stock price rising is 20%, and the likelihood of Company B's stock price rising is 30%. Assume that the returns of Company A and Company B stock are independent of each other. The probability that the stock price of at least one of the companies will rise is ______.
44%.
Let P(A)=0.3 and P(B)=0.4.P(A)=0.3 and P(B)=0.4. Suppose A and B are independent. What is the value of P(B|A)? 0.12 0.3 0.4 0.7
.4
Which of the following does not represent a continuous random variable? Multiple Choice Height of oak trees in a park. Heights and weights of newborn babies. Time of a flight between Chicago and New York. The number of customer arrivals to a bank between 10 am and 11 am.
The number of customer arrivals to a bank between 10 am and 11 am.
Which of the following is best described as a Poisson variable? Multiple Choice The number of students earning an A grade in Business Statistics. The number of heads in coin tossing. The number of ones in die casting. The number of positive reviews in a week.
The number of positive reviews in a week. The number of positive reviews in a week is best described as a Poisson variable because it deals with the number of occurrences of a certain event over time and the number of occurrences is proportional to the size of the time interval.
When some objects are randomly selected, which of the following is true? Multiple Choice The order in which objects are selected matters in combinations. The order in which objects are selected does not matter in permutations. The order in which objects are selected does not matter in combinations. The order in which objects are selected matters in both permutations and combinations.
The order in which objects are selected does not matter in combinations.
Events are collectively exhaustive if they include all events they are included in all events they contain all outcomes of an experiment they do not share any common outcomes of an experiment
they contain all outcomes of an experiment
What is the standard deviation of the number of cars sold by the salesperson during a week? Multiple Choice 0.56 0.75 0.80 1
.75 The standard deviation of the discrete random variable X is calculated as SD(X)=σ=σ2−−√.SD(X)=σ=σ2. The variance of the discrete random variable X is calculated as Var(X) = σ2 = ∑(xi − μ)2 P(X = xi).E(X) = 0 × 10/25 + 1 × 10/25 + 2 × 5/25 = 0.80Var(X) = (0 − 0.80)2 × 10/25 + (1 − 0.80)2 × 10/25 + (2 − 0.80)2 × 5/25 = 0.56SD(X)=0.56−−−−√=0.75
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. Multiple Choice 0.0000 0.5000 0.7580 0.2420
0.0000
Peter applied to an accounting firm and a consulting firm. He knows that 30% of similarly qualified applicants receive job offers from the accounting firm, while only 20% of similarly qualified applicants receive job offers from the consulting firm. Assume that receiving an offer from one firm is independent of receiving an offer from the other. What is the probability that both firms offer Peter a job?
0.06
The contingency table below provides frequencies for the preferred type of exercise for people under the age of 35, and those 35 years of age or older. Find the probability that an individual prefers biking given that he or she is 35 years old or older.
0.1698
On a particular production line, the likelihood that a light bulb is defective is 5%. Ten light bulbs are randomly selected. What are the mean and variance of the number of defective bulbs? Multiple Choice 0.475 and 0.475 0.475 and 0.6892 0.50 and 0.475 0.50 and 0.6892
0.50 and 0.475 The mean and the variance of a binomial random variable are calculated as μ = np and σ2 = npq respectively. The Excel function used is μ = 10 × 0.05 = 0.50; σ2 = 10 × 0.05 × 0.95 = 0.475
For a particular clothing store, a marketing firm finds that 16% of $10-off coupons delivered by mail are redeemed. Suppose six customers are randomly selected and are mailed $10-off coupons. What is the expected number of coupons that will be redeemed? Multiple Choice 0.81 0.96 3.42 5.04
0.96 The expected value of a binomial random variable is calculated as E(X) = μ = np. μ = 6 × 0.16 = 0.96
According to geologists, the San Francisco Bay Area experiences five earthquakes with a magnitude of 6.5 or greater every 100 years. What is the standard deviation of the number of earthquakes with a magnitude of 6.5 or greater striking the San Francisco Bay Area in the next 40 years? Multiple Choice 1.414 2.000 2.236 5.000
1.414 The standard deviation of a Poisson random variable is computed as SD(X)=σ=μ−−√.SD(X)=σ=μ.5 earthquakes/100 years relates to 2 earthquakes/40 yearsσ = 2-√2 = 1.414
According to a Department of Labor report, the city of Detroit had a 20% unemployment rate in May. Eight working-age residents were chosen at random. What was the expected number of unemployed residents, when eight working-age residents were randomly selected? Multiple Choice 1.0 1.6 2.0 6.4
1.6 The expected value of a binomial random variable is calculated as E(X) = μ = np.The Excel function used is μ = 8 × 0.20 = 1.6
The expected value is _____. rev: 09_08_2021_QC_CS-276700 Multiple Choice 0.89 0.94 1.65 1.90
1.90 The variance of the discrete random variable X is calculated asVar(X) = σ2 = ∑(xi - μ)2 P(X = xi).E(X) = 0 × 0.10 + 1 × 0.20 + 2 × 0.40 + 3 × 0.30 = 1.9Var(X) = (0 - 1.9)2 × 0.10 + (1 - 1.9)2 × 0.20 + (2 - 1.9)2 × 0.40 + (3 - 1.9)2 × 0.30 = 0.89
The height of the probability density function f(x) of the uniform distribution defined on the interval [a, b] is ______. Multiple Choice 1/(b − a) between a and b, and zero otherwise (b − a)/2 between a and b, and zero otherwise (a + b)/2 between a and b, and zero otherwise 1/(a + b) between a and b, and zero otherwise
1/(b − a) between a and b, and zero otherwise
According to geologists, the San Francisco Bay Area experiences five earthquakes with a magnitude of 6.5 or greater every 100 years. What is the expected value of the number of earthquakes with a magnitude of 6.5 or greater striking the San Francisco Bay Area in the next 40 years? Multiple Choice 1.414 2.000 2.236 5.000
2.000 The expected value of a Poisson random variable is computed as E(X) = µ. So, 5 earthquakes/100 years relates to 2 earthquakes/40 years. E(X) = 2
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. Multiple Choice 2.5% 5% 95% 97.5%
97.5%
How would you characterize a consumer who is risk loving? Multiple Choice A consumer who may accept a risky prospect even if the expected gain is negative. A consumer who demands a positive expected gain as compensation for taking risk. A consumer who completely ignores risk and makes his or her decisions solely on the basis of expected values. A consumer who is indifferent to risk.
A consumer who may accept a risky prospect even if the expected gain is negative.
Which of the following is correct? Multiple Choice A continuous random variable has a probability density function but not a cumulative distribution function. A discrete random variable has a probability mass function but not a cumulative distribution function. A continuous random variable has a probability mass function, and a discrete random variable has a probability density function. A continuous random variable has a probability density function, and a discrete random variable has a probability mass function.
A continuous random variable has a probability density function, and a discrete random variable has a probability mass function.
Which of the following statements is the most accurate about a Poisson random variable? Multiple Choice It counts the number of successes in a given number of trials. It counts the number of successes in a specified time or space interval. It is a continuous random variable. It has a bell-shaped distribution.
It counts the number of successes in a specified time or space interval.
Let A and B be two independent events with P(A) = 0.40 and P(B) = 0.20. Which of the following is correct? P(B|A)=0.40P(B|A)=0.40 P(A|B)=0.08P(A|B)=0.08 P(A∩B)=0P(A∩B)=0 P(A∪B)=0.52
P(A∪B)=0.52
Which of the following is FALSE about a continuous random variable? Multiple Choice It has uncountable value within an interval. The probability that its value is within a specific interval is equal to one. Its probability density function is nonnegative. The cumulative distribution function of a specific value of the variable is defined as the area under its probability density function up to that value.
The probability that its value is within a specific interval is equal to one.
What does it mean when we say that the tails of the normal curve are asymptotic to the x axis? Multiple Choice The tails get closer and closer to the x axis but never touch it. The tails get closer and closer to the x axis and eventually touch it. The tails get closer and closer to the x axis and eventually cross this axis. The tails get closer and closer to the x axis and eventually become this axis.
The tails get closer and closer to the x axis but never touch it.
Let X be normally distributed with mean μ and standard deviation σ > 0. Which of the following is false about the z value corresponding to a given x value? Multiple Choice A positive z = (x − μ)/σ indicates how many standard deviations x is above μ. A negative z = (x − μ)/σ indicates how many standard deviations x is below μ. The z value corresponding to x = μ is zero. The z value corresponding to a given value of x assumes any value between 0 and 1.
The z value corresponding to a given value of x assumes any value between 0 and 1.
A consumer who is risk neutral is best characterized as ______________________________________________________. Multiple Choice a consumer who may accept a risky prospect even if the expected gain is negative a consumer who demands a positive expected gain as compensation for taking risk a consumer who completely ignores risk and makes his or her decisions based solely on expected values a consumer who underplays risk
a consumer who completely ignores risk and makes his or her decisions based solely on expected values
A consumer who is risk averse is best characterized as __________. Multiple Choice a consumer who may accept a risky prospect even if the expected gain is negative a consumer who demands a positive expected gain as compensation for taking risk a consumer who completely ignores risk and makes his or her decisions based solely on expected values
a consumer who demands a positive expected gain as compensation for taking risk
