Question 1

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 presented in the problem involves a fixed number of trials (125 computers) with two possible outcomes (the computer either needs repair or it doesn't), with a constant probability of success (0.025). Therefore, we can model this situation with the binomial distribution.

To find the probability that exactly 20 of the computers will require repair on a given day, we can use the following formula for the binomial distribution:

P(X = 20) = (125 choose 20) * 0.025^20 * (1 - 0.025)^105

where (125 choose 20) represents the number of ways we can choose 20 computers out of 125, 0.025^20 is the probability of 20 successes (computers needing repair) and (1 - 0.025)^105 is the probability of 105 failures (computers not needing repair).

Note that we cannot use the Poisson distribution in this scenario because it assumes a variable rate of occurrence for an event, whereas the probability of a computer needing repair is constant.

Question 2

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 is a probability distribution for a discrete variable, which means that the variable can only take on certain values, usually integers. It is not appropriate for continuous variables.

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 determined by the weighted average of the expected returns of the individual investments in the portfolio. The covariance between the two investments affects the portfolio variance, which in turn affects the portfolio risk. However, it does not affect the portfolio expected return. Therefore, the portfolio expected return does not depend on the covariance.

Question 4

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 $, which is $ 10 \times 0.70 = 7.00 $.

Question 5

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 problem because it models the number of occurrences of rare events that happen randomly and independently over time or space, such as the number of defects in a production process. In this case, the number of blue chocolate chips per bag is rare and random, and the mean and variance are equal, which is a characteristic of the Poisson distribution. The binomial distribution would be appropriate if we were sampling with replacement and counting the number of blue chocolate chips in a fixed number of bags or trials, but this is not the case in this problem.

Question 6

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 7

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 scenario described fits the Poisson distribution, which is used to model the number of occurrences of an event in a fixed interval of time or space when the events are rare and randomly distributed. In this case, the event is receiving an e-mail from a student, and the fixed interval of time is one day. The fact that the average number of e-mails received is given (24.7) also supports the use of the Poisson distribution. The binomial distribution is used to model the number of successes in a fixed number of independent trials, such as tossing a coin or rolling dice, and does not apply to this scenario.

Question 8

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 assigns a probability of occurrence to each outcome or event.

Question 9

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

There are two equally likely outcomes for each student being chosen, either they are undergraduate or graduate. Since half of the program is undergraduate, the probability that each selected student is undergraduate is 1/2. Thus, the probability that all 4 students selected are undergraduate is (1/2)^4 = 1/16 = 0.0625, which is option B.

Question 10

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 the event of interest remains stable from trial to trial. This is because each trial has only two possible outcomes (success or failure) with a fixed probability of success (p) and failure (q). The other options are incorrect because the variable X is discrete, not continuous (A), there is no minimum requirement for the number of trials (C), and the results of one trial are independent of the results of the other trials (D).

Question 11

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 binomial distribution is used to model the number of successes in a fixed number of independent trials, each with the same probability of success. In this case, the student is answering each question randomly, and each question has 4 choices, so the probability of getting a question right by guessing is 1/4. Therefore, we can use the binomial distribution to calculate the probability of the student getting at least 20 questions right out of 30.

Question 12

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 (represented by lambda) is equal to the variance. Therefore, the mean equals the variance but not necessarily the standard deviation or median.

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

Expected value or expected gain refers to the average amount a player can expect to win or lose over many plays of a game. In the case of roulette, it takes into account the probability of each possible outcome and the amount that will be paid out for each winning bet. B is the correct answer because it refers to the amount you expect to gain in the long run over many plays, which is a more accurate reflection of a player's overall experience at the casino.

Question 14

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 15

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

A binomial distribution with an equal probability of success and failure (i.e. p=q=0.5) is always symmetric regardless of the sample size. Therefore, the given equation indicates an equal number of successes and failures, resulting in a symmetric distribution.

Question 16

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 $\pi = 0.1$ and n is small, the binomial distribution is right-skewed because the probability of success is low, leading to a distribution with a long tail on the right.

Question 17

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 situation described involves a fixed number of trials (100) and the probability of success (false claim) is known (3%). This is a classic example of a binomial distribution, where the number of successes in a fixed number of independent trials is modeled. The Poisson distribution would not be appropriate here because it is used to model the number of rare events that occur in a fixed interval of time or space, whereas the problem does not specify any time or space interval. Therefore, the correct choice is A.

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 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 19

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 occurring in a fixed interval of time or space, given the average rate of occurrence and assuming that the events occur independently and with a constant probability. In this case, we are interested in the probability of no customers arriving in a given interval of time (e.g. one minute), and the average rate of customer arrivals is given. Therefore, we can use the Poisson distribution to find the probability of no customers arriving at a checkout counter. The binomial distribution, on the other hand, is used to model the number of successes in a fixed number of trials, given a constant probability of success and assuming that the trials are independent. It is not applicable in this case since we are interested in the number of customers arriving over a continuous interval of time rather than a fixed number of trials.

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 the binomial distribution is given by $\sqrt{np(1-p)}$. Substituting the given values, we get $\sqrt{10\times0.70\times0.30} = 1.45$.

Quiz 5: Discrete Probability Distributions

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?

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

The portfolio expected return of two investments

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

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 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 probability distribution is an equation that

In a binomial distribution

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?

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

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

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 $\pi$ = 0.5, the binomial distribution will

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

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?

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