Question 1

In a binomial distribution
A)the variable X is continuous.
B)the probability of event of interest  $\pi$ is stable from trial to trial.
C)the number of trials n must be at least 30.
D)the results of one trial are dependent on the results of the other trials.

Accepted Answer

In a binomial distribution, the probability of success (or the event of interest) remains constant from trial to trial. This is a defining characteristic of the binomial distribution, which models the number of successes in a fixed number of independent trials, each with the same probability of success.

Question 2

What type of probability distribution will most likely be used to analyze warranty repair needs on new cars in the following problem?The service manager for a new automobile dealership reviewed dealership records of the past20 sales of new cars to determine the number of warranty repairs he will be called on to perform in the next 90 days.Corporate reports indicate that the probability any one of their new cars needs a warranty repair in the first 90 days is 0.05.The manager assumes that calls for warranty repair are independent of one another and is interested in predicting the number of warranty repairs he will be called on to perform in the next 90 days for this batch of 20 new cars sold.
A)binomial distribution.
B)Poisson distribution.
C)Either of the above
D)none of the above.

Accepted Answer

The problem describes a situation with a fixed number of trials (20 new cars sold), each with only two possible outcomes (a car needs a warranty repair or it does not), a constant probability of success (0.05 chance of needing a warranty repair), and the assumption that each trial is independent of the others. These characteristics match the criteria for a binomial distribution.

Question 3

Which of the following about the binomial distribution is not a true statement?
A)The probability of the event of interest must be constant from trial to trial.
B)Each outcome is independent of the other.
C)Each outcome may be classified as either "event of interest" or "not event of interest."
D)The variable of interest is continuous.

Accepted Answer

The binomial distribution deals with discrete variables, not continuous ones. Therefore, statement D is not true.

Question 4

The portfolio expected return of two investments
A)will be higher when the covariance is zero.
B)will be higher when the covariance is negative.
C)will be higher when the covariance is positive.
D)does not depend on the covariance.

Accepted Answer

The answer of The portfolio expected return of two investments
A)will...

Question 5

Whenever  $\pi$ = 0.5, the binomial distribution will
A)always be symmetric.
B)be symmetric only if n is large.
C)be right-skewed.
D)be left-skewed.

Accepted Answer

The condition F1F1F1S1πF1F1F10 = 0.5 implies that the probability of success (p) is 0.5. When p = 0.5, the binomial distribution is always symmetric, regardless of the value of n (number of trials).

Question 6

A probability distribution is an equation that
A)associates a probability of occurrence with each outcome.
B)measures outcomes and assigns values of X to the simple events.
C)assigns a value to the variability of the set of events.
D)assigns a value to the center of the set of events.

Accepted Answer

A probability distribution is a function that assigns a probability to each possible outcome of a random variable. In other words, it associates a probability of occurrence with each outcome.

Question 7

A company has 125 personal computers.The probability that any one of them will require repair on a given day is 0.025.To find the probability that exactly 20 of the computers will require repair on a given day, one will use what type of probability distribution?
A)binomial distribution.
B)Poisson distribution.
C)Either of the above
D)none of the above.

Accepted Answer

The scenario involves a fixed number of trials (125 computers), each with two possible outcomes (either a computer requires repair or it does not), and a constant probability of success (repair needed) for each trial. This fits the criteria for a binomial distribution.

Question 8

If the outcomes of a variable follow a Poisson distribution, then them
A)mean equals the standard deviation.
B)median equals the standard deviation.
C)mean equals the variance.
D)median equals the variance.

Accepted Answer

In a Poisson distribution, the mean is equal to the variance. Therefore, the answer is C. Neither the mean nor the median is equal to the standard deviation or the variance in a Poisson distribution.

Question 9

n the average, 1.8 customers per minute arrive at any one of the checkout counters of a grocery store.What type of probability distribution can be used to find out the probability that there will be no customer arriving at a checkout counter?
A)binomial distribution.
B)Poisson distribution.
C)Either of the above
D)none of the above.

Accepted Answer

The Poisson distribution is used to model the number of events (in this case, customer arrivals) in a fixed interval of time or space when these events occur with a known constant mean rate and independently of the time since the last event.

Question 10

Whenever $\pi = 0.1$ and n is small, the binomial distribution will be
A)symmetric.
B)right-skewed.
C)left-skewed.
D)None of the above.

Accepted Answer

When $\pi$ is small (less than 0.5) and n is small, the binomial distribution tends to be right-skewed because there is a higher probability of getting a small number of successes.

Question 11

If n = 10 and $\pi = 0.70$ , then the mean of the binomial distribution is
A)0.07
B)1.45.
C)7.00
D)14.29

Accepted Answer

The mean of a binomial distribution is given by $\mu = n\pi$, where $n$ is the number of trials and $\pi$ is the probability of success on a single trial. Substituting $n = 10$ and $\pi = 0.70$, we get $\mu = 10 \times 0.70 = 7.00$.

Question 12

A professor receives, on average, 24.7 e-mails from students the day before the midterm exam.To compute the probability of receiving at least 10 e-mails on such a day, he will use what type of probability distribution?
A)binomial distribution.
B)Poisson distribution.
C)Either of the above
D)none of the above.

Accepted Answer

The Poisson distribution is used to model the number of events (in this case, receiving e-mails) occurring within a fixed interval of time or space when these events happen with a known constant mean rate (24.7 e-mails per day) and independently of the time since the last event.

Question 13

The connotation "expected value" or "expected gain" from playing roulette at a casino means
A)the amount you expect to "gain" on a single play.
B)the amount you expect to "gain" in the long run over many plays.
C)the amount you need to "break even" over many plays.
D)the amount you should expect to gain if you are lucky.

Accepted Answer

The expected value or expected gain in gambling refers to the amount of money a player can expect to win (or lose) over the long run. It is calculated by multiplying each possible outcome by its probability, and then summing these products. Therefore, option B is the correct answer as it refers to the expected gain over many plays. Options A, C, and D are not correct as they refer to a single play, breaking even or luck, which are not reliable indicators of a player's long-term gains in gambling.

Question 14

A campus program evenly enrolls undergraduate and graduate students.If a random sample of 4 students is selected from the program to be interviewed about the introduction of a new fast food outlet on the ground floor of the campus building, what is the probability that all 4 students selected are undergraduate students?
A)0.0256
B)0.0625
C)0.16
D)1.00

Accepted Answer

Since the program evenly enrolls undergraduate and graduate students, the probability of selecting an undergraduate student is 0.5. Since all four students need to be undergraduate, we can use the multiplication rule of probability:
P(selecting all undergraduate students) = 0.5 x 0.5 x 0.5 x 0.5 = 0.0625 or 6.25%.
Therefore, the answer is B) 0.0625.

Question 15

What type of probability distribution will most likely be used to analyze the number of blue chocolate chips per bag in the following problem?The quality control manager of a candy plant is inspecting a batch of chocolate chip bags.When the production process is in control, the mean number of blue chocolate chips per bagis 6.0.The manager is interested in analyzing the probability that any bag being inspected has fewer than 5.0 blue chocolate chips.
A)binomial distribution.
B)Poisson distribution.
C)Either of the above
D)none of the above.

Accepted Answer

The Poisson distribution is most appropriate for this scenario because it is used to model the number of times an event happens in a fixed interval of time or space, and it deals well with rare events (like finding a specific number of blue chips in a bag). Given the context, where we're looking at an average occurrence (mean) of a specific event (blue chips per bag), and interested in the probability of a certain number of these events occurring, the Poisson distribution fits the criteria.

Question 16

A multiple-choice test has 30 questions.There are 4 choices for each question.A student who has not studied for the test decides to answer all questions randomly.What type of probability distribution can be used to figure out his chance of getting at least 20 questions right?
A)binomial distribution.
B)Poisson distribution.
C)Either of the above
D)none of the above.

Accepted Answer

The situation described fits a binomial distribution because there are a fixed number of trials (30 questions), each trial has only two possible outcomes (correct or incorrect), the probability of success (getting a question right) is constant (1/4), and the trials are independent.

Question 17

Thirty-six of the staff of 80 teachers at a local intermediate school are certified in Cardio- Pulmonary Resuscitation (CPR).In 180 days of school, about how many days can we expect that the teacher on bus duty will likely be certified in CPR?
A)5 days
B)45 days
C)65 days
D)81 days

Accepted Answer

The probability that a teacher on bus duty is certified in CPR is 36/80, or 45%. Over 180 days, we can expect about 45% of those days to have a CPR-certified teacher on bus duty, which is 0.45 * 180 = 81 days.

Question 18

What type of probability distribution will the consulting firm most likely employ to analyze the insurance claims in the following problem?An insurance company has called a consulting firm to determine if the company has an unusually high number of false insurance claims.It is known that the industry proportion for false claims is 3%.The consulting firm has decided to randomly and independently sample100 of the company's insurance claims.They believe the number of these 100 that are false will yield the information the company desires.
A)binomial distribution.
B)Poisson distribution.
C)Either of the above
D)none of the above.

Accepted Answer

The scenario involves a fixed number of trials (100 insurance claims), each with two possible outcomes (false or not false), and a constant probability of success (the industry proportion for false claims is 3%). These conditions fit the criteria for a binomial distribution.

Question 19

A lab orders 100 rats a week for each of the 52 weeks in the year for experiments that the lab conducts.Suppose the mean cost of rats used in lab experiments turned out to be $13.00 per week.Interpret this value.
A)Most of the weeks resulted in rat costs of $13.00.
B)The median cost for the distribution of rat costs is $13.00.
C)The expected or mean cost for all weekly rat purchases is $13.00.
D)The rat cost that occurs more often than any other is $13.00.

Accepted Answer

The mean cost of rats used in lab experiments being $13.00 per week means that if you were to take the total cost of rats used in lab experiments for the year (52 weeks x 100 rats per week = 5200 rats), and divide it by 52 (the number of weeks in a year), the result would be $13.00. This is the expected or mean cost for all weekly rat purchases.

Question 20

If n = 10 and $\pi = 0.70$ , then the standard deviation of the binomial distribution is
A)0.07
B)1.45
C)7.00
D)14.29

Accepted Answer

The standard deviation of a binomial distribution is given by the formula $\sqrt{n\pi(1-\pi)}$, where $n$ is the number of trials and $\pi$ is the probability of success. Substituting $n = 10$ and $\pi = 0.70$, we get $\sqrt{10*0.70*(1-0.70)} = \sqrt{10*0.70*0.30} = \sqrt{2.1} \approx 1.45$.

Quiz 5: Discrete Probability Distributions

In a binomial distribution

Which of the following about the binomial distribution is not a true statement?

The portfolio expected return of two investments

Whenever $\pi$ = 0.5, the binomial distribution will

A probability distribution is an equation that

A company has 125 personal computers.The probability that any one of them will require repair on a given day is 0.025.To find the probability that exactly 20 of the computers will require repair on a given day, one will use what type of probability distribution?

If the outcomes of a variable follow a Poisson distribution, then them

n the average, 1.8 customers per minute arrive at any one of the checkout counters of a grocery store.What type of probability distribution can be used to find out the probability that there will be no customer arriving at a checkout counter?

Whenever $\pi = 0.1$ and n is small, the binomial distribution will be

If n = 10 and $\pi = 0.70$ , then the mean of the binomial distribution is

A professor receives, on average, 24.7 e-mails from students the day before the midterm exam.To compute the probability of receiving at least 10 e-mails on such a day, he will use what type of probability distribution?

The connotation "expected value" or "expected gain" from playing roulette at a casino means

A multiple-choice test has 30 questions.There are 4 choices for each question.A student who has not studied for the test decides to answer all questions randomly.What type of probability distribution can be used to figure out his chance of getting at least 20 questions right?

Thirty-six of the staff of 80 teachers at a local intermediate school are certified in Cardio- Pulmonary Resuscitation (CPR) .In 180 days of school, about how many days can we expect that the teacher on bus duty will likely be certified in CPR?

A lab orders 100 rats a week for each of the 52 weeks in the year for experiments that the lab conducts.Suppose the mean cost of rats used in lab experiments turned out to be $13.00 per week.Interpret this value.

If n = 10 and $\pi = 0.70$ , then the standard deviation of the binomial distribution is