Question 1

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 situation described fits the criteria for a binomial distribution, which is used when there are a fixed number of independent trials with only two outcomes (success or failure) and a constant probability of success. In this case, answering a question correctly is a success or a failure with a probability of 0.25 for each option. The question asks for the probability of getting at least 20 questions right, which can be calculated using the binomial distribution formula.

Question 2

If  $\pi$ 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 = n*p, where n is the number of trials and p is the probability of success. Therefore, if n increases, the mean will also increase, assuming p remains constant.

Question 3

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 a Poisson distribution represents the average or expected number of events occurring in a given time period. In this case, the mean of 6.5 tickets per day means that if we sampled all days, we would expect the average number of tickets written to be 6.5 per day.

Question 4

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 probability of a certain number of events occurring in a fixed interval of time or space, given a known average rate of occurrence. In this case, the arrival of customers at the checkout counter of the grocery store follows a Poisson distribution since the rate of arrival is known (1.8 customers per minute) and the number of arrivals in a given interval of time (e.g. a minute) can vary. The probability of no customer arriving at a checkout counter can be found using the Poisson distribution with a mean of λ = 1.8 and a value of x = 0, where x represents the number of events (in this case, customers arriving) that we want to find the probability of.

Question 5

The Poisson distribution can be used to model a continuous random variable.

Accepted Answer

The Poisson distribution is a discrete probability distribution and can only be used to model discrete random variables.

Question 6

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

Accepted Answer

A binomial distribution requires that each trial has only two possible outcomes (success or failure) and the probability of success remains constant from trial to trial. In the case of customers arriving at a department store, there are more than two possible outcomes (e.g. some may buy something while others may not), and the probability of a customer arriving may not be constant over time (e.g. it may be higher during certain times of day or on certain days of the week). Therefore, a binomial distribution would not be appropriate to model this situation.

Question 7

The diameters of 10 randomly selected bolts have a binomial distribution.

Accepted Answer

A binomial distribution represents the number of successes in a fixed number of independent trials with only two possible outcomes (success or failure). The diameter of a bolt is a continuous variable and not a binary outcome, so it cannot be modeled using a binomial distribution.

Question 8

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

Let X be the number of incorrect verdicts in 10 decisions. Then X follows a binomial distribution with n=10 and p=0.1.

We want to find P(X<2) = P(X=0) + P(X=1). Using the binomial probability formula, we have:

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

Therefore, P(X<2) = P(X=0) + P(X=1) = 0.3487 + 0.3874 = 0.7361, which is approximately 0.736.

So the statement is true.

Question 9

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 10

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 hypergeometric distribution is appropriate in this scenario because it is used to model the probability of obtaining a certain number of successes when sampling without replacement from a finite population with a specific number of successes and failures. In this case, the analyst is selecting 3 companies from a population of 60, with only 3 of them being highly successful. The binomial distribution would be appropriate if the analyst was selecting companies with replacement, and the Poisson distribution is used to model the number of occurrences of an event in a fixed interval of time or space, which is not applicable here.

Question 11

Another name for the mean of a probability distribution is its expected value.

Accepted Answer

This is true. The expected value is calculated as the sum of each possible outcome multiplied by its probability, which is the same as the calculation for the mean.

Question 12

In a Poisson distribution,the mean and variance are equal.

Accepted Answer

This is a characteristic of the Poisson distribution, where both the mean and variance are equal to λ, the rate parameter.

Question 13

The largest value that a Poisson random variable X can have is n.

Accepted Answer

A Poisson random variable can take any non-negative integer value from 0 to infinity, meaning there is no maximum value of n.

Question 14

If  $\pi$ remains constant in a binomial distribution,an increase in n will increase the variance.

Accepted Answer

In a binomial distribution with F1F1F1S1F1F1F10 constant, the variance (σ²) is given by σ² = np(1-p), where n is the sample size and p is the probability of success. As n increases, the value of np increases, which means that the variance will also increase. Therefore, the statement is true.

Question 15

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

In a hypergeometric distribution, the sample is taken without replacement from a finite population of known size, which fits the given scenario. The number of successes (males) in the sample follows a hypergeometric distribution, making option A true.

Question 16

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

Accepted Answer

The Poisson distribution is commonly used to model the number of arrivals in a given time interval, such as customers arriving at a store within a 5-minute period. Therefore, the statement is true.

Question 17

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 18

In a Poisson distribution,the mean and standard deviation are equal.

Accepted Answer

The answer of In a Poisson distribution,the mean and standard...

Question 19

Suppose that the number of airplanes arriving at an airport per minute is a Poisson process.The mean 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 observing exactly $k$ events in a fixed interval in a Poisson process is given by $P(k; \lambda) = \frac{e^{-\lambda} \lambda^k}{k!}$, where $\lambda$ is the average rate (mean) of occurrences in that interval, and $k$ is the number of occurrences. For $\lambda = 3$ and $k = 6$, the formula yields $P(6; 3) = \frac{e^{-3} 3^6}{6!} \approx 0.0504$, matching the provided probability.

Question 20

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

A binomial distribution requires fixed and known probabilities of success and failure for each trial, which is not the case in this scenario since the probability of selecting a male changes with each selection without replacement. Therefore, the number of males selected does not have a binomial distribution.

Quiz 5: Discrete Probability Distributions

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 π\piπ remains constant in a binomial distribution,an increase in n will not change the mean.

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.

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?