Question 1

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 be positive or negative, depending on the direction of the relationship between the two variables. It can take on any value between negative infinity and positive infinity. Therefore, options A, B, and D are incorrect.

Question 2

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 calculated as $ n \times \pi $. Here, $ n = 10 $ and $ \pi = 0.70 $, so the mean is $ 10 \times 0.70 = 7.00 $.

Question 3

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 portfolio expected return is calculated as the weighted average of the expected returns of the individual investments and does not depend on the covariance between them. Covariance affects the portfolio's risk, not its expected return.

Question 4

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.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 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 hypergeometric distribution, which models the probability of obtaining a specific number of successes (in this case, stocks that appreciate more than 20%) in a fixed sample size (3 stocks) without replacement from a finite population (25 stocks). The binomial distribution would apply if the analyst was picking with replacement from an infinite population or a very large population. The Poisson distribution would apply if the events were rare and could be approximated by a Poisson process, but this is not the case here.

Question 5

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 formula for the standard deviation of a binomial distribution with n trials and probability of success p is:
σ = √(np(1-p))
Given n = 10 and F1F1F1S1F1F1F10 = 0.70, we can solve for p:
0.70 = (6/10)p + (4/10)(1-p)
0.70 = 0.6p + 0.4 - 0.4p
0.3 = 0.2p
p = 1.5
This cannot be the true probability of success since it is greater than 1. Therefore, we made an error in calculating p. We can find the correct value of p by rearranging the equation:
(6/10)p = 0.70 - (4/10)(1-p)
0.6p = 0.3 + 0.4p
0.1p = 0.3
p = 3
Again, this cannot be the true probability of success since it is greater than 1. Therefore, there is no solution to this equation and the given information is impossible. We cannot calculate the standard deviation under these conditions.

Question 6

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 appropriate for calculating the probability of rare events occurring in a fixed interval of time or space. In this case, the professor is interested in the number of emails received in a day, and the average is given, making Poisson a suitable choice. The binomial and hypergeometric distributions are used for different types of problems, such as counting successes in multiple independent trials or sampling without replacement, respectively, and are not appropriate here.

Question 7

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 equals the variance. This is a characteristic property of Poisson distributions. However, the standard deviation is equal to the square root of the variance, so it is not necessarily equal to the mean or median.

Question 8

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

The binomial distribution becomes right-skewed when the probability of success is small (in this case, F1F1F1S1). As given in the question, F1F1F1S1F1F1F10=0.1 which indicates a small probability of success. Hence, the binomial distribution will be right-skewed.

Question 9

In a binomial distribution
A)the variable X is continuous.
B)the probability of event of interest 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 variable X is discrete, not continuous. The probability of the event of interest is stable from trial to trial, which is why it is called a "success probability." There is no specific requirement for the number of trials n to be at least 30. The results of one trial are independent of the results of the other trials, which is what makes this distribution binomial. Therefore, the best choice is B.

Question 10

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 11

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.

Question 12

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 enrolls undergraduate and graduate students evenly, the probability of selecting an undergraduate student is 1/2. For 4 students selected independently, the probability that all are undergraduates is (1/2)^4 = 1/16 = 0.0625.

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 roulette refers to the average amount a person can expect to win or lose over many plays. It is calculated based on the probabilities of winning or losing and takes into account the house edge. It is not the amount you expect to gain on a single play or if you are lucky. It also does not refer to the amount needed to break even over many plays, as this would be the expected value or expected gain of 0.

Question 14

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 bag is 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)hypergeometric distribution.
D)none of the above.

Accepted Answer

The Poisson distribution is typically used to analyze the probability of rare events occurring in a fixed period of time or space. In this case, we are analyzing the probability of having fewer than 5 blue chocolate chips in a single bag, which can be considered a rare event. Additionally, the mean number of blue chocolate chips per bag is given, which is a characteristic of the Poisson distribution.

Question 15

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 described fits the description of a binomial distribution, where there are a fixed number of trials (125 computers), each trial can result in one of two outcomes (require repair or not require repair), the probability of success (requiring repair) is constant (0.025), and the trials are independent of each other. The question specifically asks for the probability of exactly 20 of the computers requiring repair, which is the type of scenario that can be solved using the binomial distribution formula.

Question 16

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 situation described involves a binomial distribution, as it involves a fixed sample size (100) and independent trials (checking each of the 100 insurance claims individually) with a probability of success (3% false claims).

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 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 18

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 scenario fits the criteria for a binomial distribution because there are a fixed number of trials (20 cars), each trial has only two possible outcomes (a car needs a warranty repair or it doesn't), the probability of needing a repair is constant (0.05), and each trial is independent of the others.

Question 19

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 problem involves randomly selecting a sample of 8 cars from a population of 48, which makes the situation best suited for a hypergeometric distribution. This distribution is used for situations where the sample size is small and the population size is limited, and the number of successes and failures in the sample is of interest. The binomial distribution would be appropriate if we were randomly selecting individual cars from the population and checking for defective radios, but not when selecting a sample without replacement. The Poisson distribution is used for counting the occurrence of rare events over a continuous time period, and is not applicable in this situation.

Question 20

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 associates a probability of occurrence with each possible outcome. This allows us to understand the likelihood of each outcome, which is important for making decisions and predictions in various fields such as statistics, finance, and engineering.

Quiz 5: Discrete Probability Distributions

The covariance

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

The portfolio expected return of two investments

If n = 10 and $\pi$ = 0.70,then the standard deviation 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?

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

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

In a binomial distribution

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

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

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

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?

A probability distribution is an equation that