Question 1

Whenever π = 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

When π = 0.5, the binomial distribution is symmetric for any value of n. This is because the probability of success and failure are equal, resulting in a balanced distribution.

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 past 20 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) Hypergeometric distribution
D) None of the above.

Accepted Answer

The problem involves a fixed number of trials (20 cars sold), each of which can either result in a warranty repair or not (success/failure). The probability of success (needing a warranty repair) is constant (0.05) for each trial, and the trials are independent of each other. These conditions indicate the use of a binomial distribution.

Question 3

In a binomial distribution
A) the random variable X is continuous.
B) the probability of event of interest π 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

This is because in a binomial distribution, the probability of the event of interest (π) remains constant from trial to trial. This is a defining characteristic of a binomial distribution. The other options are incorrect because the random variable X is discrete, not continuous (option A), the number of trials n can be any positive integer (option C), and the results of each trial are independent of the results of the other trials (option D).

Question 4

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) Hypergeometric distribution
D) None of the above.

Accepted Answer

The Poisson distribution is the best choice for this scenario since it models events that occur randomly over a continuous interval of time or space, such as the number of emails received in a day. The binomial distribution models events that have a fixed number of trials with a binary outcome (e.g. flipping a coin), while the hypergeometric distribution models events in a finite population without replacement (e.g. drawing cards from a deck). Therefore, neither the binomial nor hypergeometric distribution would be appropriate for the scenario described.

Question 5

A stock analyst was provided with a list of 25 stocks. He was expected to pick 3 stocks from the list whose prices are expected to rise by more than 20% after 30 days. In reality, the prices of only 5 stocks would rise by more than 20% after 30 days. If he randomly selected 3 stocks from the list, he would use what type of probability distribution to compute the probability that all of the chosen stocks would appreciate more than 20% after 30 days?
A) Binomial distribution
B) Poisson distribution
C) Hypergeometric distribution
D) None of the above.

Accepted Answer

The situation described fits the criteria for a hypergeometric distribution, where the probability of success (i.e. choosing a stock that will appreciate by more than 20%) changes as each stock is selected without replacement. The binomial distribution is not appropriate since the probability of success remains the same with each trial, and the Poisson distribution is not appropriate since the number of successes (i.e. stocks expected to appreciate by more than 20%) is not a result of a rare event occurring at a constant rate.

Question 6

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 average number of blue chocolate chips per bag is 6.0. The manager is interested in analyzing the probability that any particular bag being inspected has fewer than 5.0 blue chocolate chips.
A) Binomial distribution
B) Poisson distribution
C) Hypergeometric distribution
D) None of the above.

Accepted Answer

The Poisson distribution would be the best choice for analyzing the number of blue chocolate chips per bag in this scenario because it is used to model the number of occurrences of a rare event in a fixed interval of time or space. In this case, the occurrence is the number of blue chocolate chips in a bag, and the interval is the bag itself. The fact that the average number of blue chocolate chips per bag is given as 6.0 also supports the use of a Poisson distribution.

Question 7

If n = 10 and π = 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 formula for the standard deviation of a binomial distribution is given by σ = √(np(1-p)), where n is the number of trials and p is the probability of success. 
Substituting n = 10 and π = 0.70, we get
σ = √(10 × 0.70 × 0.30) 
σ = √2.1 
σ ≈ 1.45

Therefore, the best choice is B.

Question 8

What type of probability distribution will most likely be used to analyze the number of cars with defective radios in the following problem? From an inventory of 48 new cars being shipped to local dealerships, corporate reports indicate that 12 have defective radios installed. The sales manager of one dealership wants to predict the probability out of the 8 new cars it just received that, when each is tested, no more than 2 of the cars have defective radios.
A) Binomial distribution
B) Poisson distribution
C) Hypergeometric distribution
D) None of the above.

Accepted Answer

The hypergeometric distribution is appropriate here because the problem involves sampling without replacement from a finite population (48 cars), where there are a fixed number of successes (12 cars with defective radios).

Question 9

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) Hypergeometric distribution
D) None of the above.

Accepted Answer

The scenario fits the criteria for a binomial distribution, which involves a fixed number of trials (125 computers), each with only two possible outcomes (require repair or not), and a constant probability of success (0.025). Therefore, the probability of exactly 20 computers requiring repair can be calculated using the binomial probability formula.

Question 10

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 proportion of teachers certified in CPR is 36 out of 80, which simplifies to 9 out of 20. To find out how many days out of 180 a CPR-certified teacher would likely be on bus duty, multiply the total days by this proportion: $180 \times \frac{9}{20} = 81$ days.

Question 11

Whenever π = 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 π = 0.1 and n is small, the distribution is right-skewed because the probability of success is low, leading to a higher concentration of probabilities at the lower end.

Question 12

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 sample 100 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) Hypergeometric distribution
D) None of the above.

Accepted Answer

The consulting firm is interested in analyzing the proportion of false insurance claims in a sample of 100 claims from the insurance company. This type of problem can be modeled using a binomial distribution, which is suitable for analyzing the number of successes (or failures) in a fixed number of independent trials, where each trial has the same probability of success. In this case, the probability of a false claim in the insurance industry is known (3%), and the sample size is fixed (100). Therefore, the binomial distribution is the most appropriate choice for analyzing this problem.

Question 13

If n = 10 and π = 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 calculated as n * π, which is 10 * 0.70 = 7.00.

Question 14

If the outcomes of a random variable follow a Poisson distribution, then their
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 and the variance are equal. Therefore, the mean also equals the variance.

Question 15

The covariance
A) must be between -1 and +1.
B) must be positive.
C) can be positive or negative.
D) must be less than +1.

Accepted Answer

The covariance can take on both positive and negative values, indicating the direction of the relationship between two variables. It can also be equal to zero, indicating no relationship between the variables. It is not limited to being less than +1, and it can be negative as well as positive. However, it is not necessarily always positive.

Question 16

A probability distribution is an equation that
A) associates a particular 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 assigns a particular probability of occurrence with each outcome.

Question 17

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 roulette refers to the amount you can expect to gain in the long run over many plays. It takes into consideration the probability of winning and losing, as well as the payout for each bet. This is important for gamblers to understand as it helps them make informed decisions about their betting strategies and manage their bankrolls effectively.

Question 18

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 half of the students in the program are undergraduates, the probability of selecting an undergraduate is 0.5. We want to find the probability that all 4 students selected are undergraduate, which is a binomial distribution with n=4 and p=0.5. 
P(X=4) = (4 choose 4) * 0.5^4 * 0.5^0 = 0.0625

Question 19

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)...

Question 20

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 random variable of interest is continuous.

Accepted Answer

The binomial distribution is a discrete probability distribution, meaning that the random variable of interest can only take on a finite or countably infinite number of values (i.e. integers). Therefore, statement D is not true since it implies that the random variable is continuous. The other statements (A, B, and C) are true and are characteristics of the binomial distribution.

Quiz 5: Discrete Probability Distributions

Whenever π = 0.5, the binomial distribution will

In a binomial distribution

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?

If n = 10 and π = 0.70, then the standard deviation of the binomial distribution is

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?

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?

Whenever π = 0.1 and n is small, the binomial distribution will be

If n = 10 and π = 0.70, then the mean of the binomial distribution is

If the outcomes of a random variable follow a Poisson distribution, then their

The covariance

A probability distribution is an equation that

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

The portfolio expected return of two investments

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