Question 1

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?

Accepted Answer

D

Question 2

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

Accepted Answer

The binomial distribution assumes a discrete random variable that follows a specific set of criteria, including a fixed number of trials, a constant probability of success, and independent events. Therefore, the random variable of interest in the binomial distribution is not continuous.

Question 3

In a binomial distribution,

Accepted Answer

In a binomial distribution, the probability of the event of interest π is stable from trial to trial. The random variable X is discrete, not continuous. There is no minimum number of trials required for a binomial distribution. The results of each trial are independent of one another.

Question 4

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.

Accepted Answer

The situation described involves a fixed number of independent trials (100 insurance claims) with a binary outcome (true or false). This fits the criteria for a binomial distribution.

Question 5

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

Accepted Answer

The answer of If n = 10 and π =...

Question 6

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

Accepted Answer

The formula for the standard deviation of a binomial distribution is √(nπ(1-π)). 
So, substituting n=10 and π=0.70, we get √(10*0.70*(1-0.70)) = 1.45. 
Therefore, the standard deviation is 1.45 which is option B.

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?

Accepted Answer

The scenario presented follows the conditions for a binomial distribution, as it involves a fixed number of trials (125 personal computers), a binary outcome (require repair or not), independence between trials (one computer's outcome does not affect the outcome of another), 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. The Poisson distribution could be used if the problem asked for the probability of a certain number of computer repairs over a given time interval, but that is not the case here.

Question 8

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

Accepted Answer

The expected value or expected gain in roulette refers to the average amount a player can expect to win or lose over a large number of bets. It is calculated by multiplying the potential winnings or losses of each bet by their respective probabilities of occurring and summing all these values. Therefore, option B is the correct choice as it refers to the expected gain over many plays.

Question 9

Whenever π = 0.5, the binomial distribution will

Accepted Answer

When π = 0.5, the probability of success and failure are equal. In such cases, the binomial distribution will always be symmetric.

Question 10

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.

Accepted Answer

The Poisson distribution is most appropriate for this problem because it is used to model the number of events occurring in a fixed interval of time or space when the events are rare and independent. In this case, the number of blue chocolate chips per bag is a rare event that can be modeled using the Poisson distribution with the average rate of 6.0 blue chocolate chips per bag. The binomial distribution would be used if the problem involved a fixed number of trials with a binary outcome (e.g., success or failure) and constant probability of success.

Question 11

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

Accepted Answer

In a Poisson distribution, the mean (𝜇) equals the variance (𝜎²). Therefore, the mean equals the square root of the variance, which is not equal to the standard deviation. The median of a Poisson distribution is not necessarily equal to the mean or the variance.

Question 12

A probability distribution is an equation that

Accepted Answer

A probability distribution assigns a particular probability of occurrence to each outcome.

Question 13

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

Accepted Answer

When π = 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 longer tail on the right.

Question 14

On 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?

Accepted Answer

The Poisson distribution is used to model the number of events occurring in a fixed interval of time or space, given a known average rate of occurrence. In this case, the average rate of arrival of customers is given (1.8 per minute) and we are interested in the probability of no customers arriving at a checkout counter in a given minute. Therefore, the Poisson distribution is the appropriate choice for this scenario. The binomial distribution would be used for a scenario where there are a fixed number of trials with a binary outcome (e.g. success/failure) and a known probability of success for each trial.

Question 15

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.

Accepted Answer

The binomial distribution is used to model the number of successes in a sequence of independent experiments, each of which has a constant probability of success. In this case, the experiments are the 20 new cars sold, and the success is a warranty repair. The probability of success is 0.05, and the number of experiments is 20. Therefore, the binomial distribution is the most likely distribution to be used to analyze warranty repair needs on new cars in this problem.

Question 16

A lab orders 100 rats a week for each of the 52 weeks in the year for experiments that the lab conducts. Prices for 100 rats follow the following distribution

Accepted Answer

To find the total cost for the year, we multiply the weekly cost by the number of weeks in a year.
$520*52 = $27040
$637*52 = $33124
$650*52 = $33800
$780*52 = $40560

Out of the given options, option B has the lowest cost for the year at $33,124.

Question 17

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?

Accepted Answer

A binomial distribution can be used in this scenario because there are a fixed number of independent trials (answering each question) with only two possible outcomes (getting the question right or wrong), and the probability of success (getting the question right) is the same for each trial (1/4).

Question 18

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?

Accepted Answer

The scenario described fits a Poisson distribution, which is used to model the number of events occurring in a fixed interval of time or space, given a known average rate of occurrence. The binomial distribution, on the other hand, is used to model the number of successes in a fixed number of independent trials, each with the same probability of success. Therefore, it is not applicable in this case, since the number of e-mails received is not the result of a series of independent trials.

Question 19

A campus program evenly enrolls undergraduate and graduate students. If a random sample of four 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 four students selected are undergraduate students?

Accepted Answer

Since the program evenly enrolls undergraduate and graduate students, the probability of selecting an undergraduate student is 0.5.

Thus, the probability of selecting four undergraduate students in a row is 0.5 * 0.5 * 0.5 * 0.5, which equals 0.0625 or 6.25%.

Therefore, the correct answer is choice B.

Question 20

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.

Accepted Answer

The mean cost of $13.00 represents the expected or average cost for all weekly rat purchases.

Quiz 5: Discrete Probability Distributions

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?