Question 1

True or False: The Poisson distribution can be used to model a continuous random variable.

Accepted Answer

The Poisson distribution is used to model discrete random variables, not continuous random variables.

Question 2

True or False: Suppose that a judge's decisions follow a binomial distribution and that his verdict is incorrect 10% of the time.In his next 10 decisions,the probability that he makes fewer than 2 incorrect verdicts is 0.736.

Accepted Answer

The probability of making an incorrect verdict is 0.1, so the probability of making a correct verdict is 0.9. We want to find the probability of making fewer than 2 incorrect verdicts in 10 decisions, which can be calculated using the binomial distribution:

P(X<2) = P(X=0) + P(X=1)
P(X=0) = (0.9)^10 * (1 choose 0) = 0.3487
P(X=1) = 10 * 0.1 * (0.9)^9 = 0.3874
P(X<2) = 0.3487 + 0.3874 = 0.7361

Therefore, the statement is true and the answer is A.

Question 3

True or False: In a Poisson distribution,the mean and variance are equal.

Accepted Answer

In a Poisson distribution, the mean and variance are equal, and they are both equal to the parameter λ, which represents the average rate of occurrence.

Question 4

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:  How much should the lab budget for next year's rat orders be,assuming this distribution does not change?
A)$520
B)$637
C)$650
D)$780

Accepted Answer

The answer of A lab orders 100 rats a week...

Question 5

True or False: Another name for the mean of a probability distribution is its expected value.

Accepted Answer

The mean of a probability distribution is also called its expected value.

Question 6

True or False: The number of customers arriving at a department store in a 5-minute period has a Poisson distribution.

Accepted Answer

The Poisson distribution is often used to model the number of rare events occurring in a fixed time or space interval. The arrival of customers at a department store can be seen as a rare event, and thus the number of customers arriving in a 5-minute period can be approximated by a Poisson distribution.

Question 7

True or False: The diameters of 10 randomly selected bolts have a binomial distribution.

Accepted Answer

The diameters of bolts are continuous values and not a discrete random variable with only two possible outcomes. Therefore, they cannot have a binomial distribution.

Question 8

True or False: Suppose that the number of airplanes arriving at an airport per minute is a Poisson process.The average number of airplanes arriving per minute is 3.The probability that exactly 6 planes arrive in the next minute is 0.05041.

Accepted Answer

The probability of exactly k events in a Poisson distribution is given by $ P(k; \lambda) = \frac{e^{-\lambda} \lambda^k}{k!} $. For k=6 and $\lambda=3$, this calculates to approximately 0.05041.

Question 9

True or False: The largest value that a Poisson random variable X can have is n.

Accepted Answer

A Poisson random variable can take on any non-negative integer value, meaning that it could potentially be infinite. Therefore, there is no largest value that a Poisson random variable can have.

Question 10

True or False: The number of customers arriving at a department store in a 5-minute period has a binomial distribution.

Accepted Answer

The number of customers arriving at a department store in a 5-minute period is not a binomial distribution because the trials (arrivals) are not independent and the probability of success (a customer arriving) is not constant. The number of arrivals may also depend on external factors, such as time of day or weather, making it difficult to model as a binomial distribution.

Question 11

True or False: The number of males selected in a sample of 5 students taken without replacement from a class of 9 females and 18 males has a binomial distribution.

Accepted Answer

The number of males selected does not have a binomial distribution because the probability of selecting a male changes with each selection due to the lack of replacement.

Question 12

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?
A)Binomial distribution
B)Poisson distribution
C)Hypergeometric distribution
D)None of the above.

Accepted Answer

The Poisson distribution is used to model the number of events occurring in a fixed amount of time when the events occur independently and at a constant rate. In this case, the arrival of customers at the checkout counters of the grocery store is an example of such an event. The probability that there will be no customer arriving at a checkout counter can be calculated using the Poisson distribution. Therefore, the best choice is B. The other options are not applicable in this scenario.

Question 13

A financial analyst is presented with information on the past records of 60 start-up companies and told that in fact only 3 of them have managed to become highly successful.He selected 3 companies from this group as the candidates for success.To analyze his ability to spot the companies that will eventually become highly successful,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 scenario described fits the Hypergeometric distribution, which deals with calculating probabilities from a finite population without replacement. In this case, the analyst is selecting 3 companies from a group of 60, and the probability of success depends on how many successful companies are actually in the group, which is a fixed population without replacement. Therefore, the Hypergeometric distribution is the most appropriate choice.

Question 14

True or False: If remains constant in a binomial distribution,an increase in n will increase the variance.

Accepted Answer

The variance of a binomial distribution is given by σ² = np(1-p), where n is the number of trials and p is the probability of success. If p remains constant and n increases, then np increases as well. Since (1-p) remains constant, an increase in np will result in an increase in the variance. Therefore, the statement is true.

Question 15

The local police department must write,on average,5 tickets a day to keep department revenues at budgeted levels.Suppose the number of tickets written per day follows a Poisson distribution with a mean of 6.5 tickets per day.Interpret the value of the mean.
A)The number of tickets that is written most often is 6.5 tickets per day.
B)Half of the days have less than 6.5 tickets written and half of the days have more than 6.5 tickets written.
C)If we sampled all days,the arithmetic average or expected number of tickets written would be 6.5 tickets per day.
D)The mean has no interpretation since 0.5 ticket can never be written.

Accepted Answer

The mean of 6.5 tickets per day represents the expected value or average number of tickets that would be written if we sampled all days.

Question 16

True or False: The number of males selected in a sample of 5 students taken without replacement from a class of 9 females and 18 males has a hypergeometric distribution.

Accepted Answer

A hypergeometric distribution is used when sampling without replacement from a finite population of two types, and the number of successes (in this case, the number of males) is of interest. Therefore, the number of males selected in this scenario follows a hypergeometric distribution.

Question 17

True or False: If remains constant in a binomial distribution,an increase in n will not change the mean.

Accepted Answer

In a binomial distribution, the mean is given by μ = nπ, where n is the number of trials and π is the probability of success. Therefore, if n increases and π remains constant, the mean will also increase.

Question 18

True or False: In a Poisson distribution,the mean and standard deviation are equal.

Accepted Answer

The answer of True or False: In a Poisson distribution,the...

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 average 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 given mean cost of rats used in lab experiments is $13.00 per week, which represents the expected or average cost for all weekly rat purchases.

Question 20

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)Hypergeometric distribution
D)None of the above.

Accepted Answer

The scenario described fits the definition of a binomial distribution: a fixed number of independent trials (answering each question), each with a fixed probability of success (getting the answer right), and the goal of finding the probability of a certain number of successes (getting at least 20 questions right).

Quiz 5: Discrete Probability Distributions