Question 1

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 2

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 independent trials with two possible outcomes and a fixed probability of success. In this situation, the probability of selecting a male changes with each selection and is dependent on the previous selections. Therefore, the number of males selected does not have a binomial distribution.

Question 3

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

Accepted Answer

The diameters of bolts are typically continuous data and not a discrete set of values that would be appropriate for a binomial distribution. Therefore, the statement is false.

Question 4

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

The hypergeometric distribution applies to situations where a sample is taken without replacement from a finite population, where the proportion of successes in the population is known. In this case, the population is the class of students, and the successes are the males. The sample size is 5, and each selection affects the probability of selecting a male on the subsequent selections. Therefore, the number of males selected in the sample has a hypergeometric distribution.

Question 5

If π remains constant in a binomial distribution,an increase in n will increase the variance.

Accepted Answer

The variance of a binomial distribution is calculated as nπ(1-π). If π remains constant and n increases, the value of nπ(1-π) also increases, leading to an increase in variance.

Question 6

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

Accepted Answer

This statement is true. The mean of a probability distribution is calculated as the weighted average of all possible values, with each value weighted by its corresponding probability. This is also known as the expected value of the distribution.

Question 7

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

Accepted Answer

The Poisson distribution is used to model a discrete random variable that represents the number of events occurring in a fixed time or space interval. It cannot be used for continuous random variables, which have an infinite number of possible values within a given range.

Question 8

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 occurrences of an event in a given time interval. In this case, the event is the arrival of customers at a checkout counter and the time interval is one minute. The Poisson distribution is appropriate because it models the occurrence of rare events (i.e. the arrival of customers) over a continuous interval of time. The probability of no customers arriving at a checkout counter can be calculated using the Poisson distribution with a mean of 1.8 customers per minute.

Question 9

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

Accepted Answer

This is true. In a Poisson distribution, the mean (λ) and variance (σ^2) are both equal to λ.

Question 10

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 in the given problem is a classic example of binomial distribution where there are multiple independent trials (answering each question) with a fixed probability (1/4 chance of getting a question right) of success (getting a question right). The question asks for the probability of getting at least 20 questions right, which can be calculated using the binomial distribution formula. Therefore, the best choice would be A) Binomial distribution.

Question 11

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: Price: $10.00 $12.50 $15.00
Probability: 0.35 0.40 0.25
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 expected cost of each order can be calculated by taking a weighted average of the prices:
Expected cost = (0.35 x $10.00) + (0.40 x $12.50) + (0.25 x $15.00) = $12.125
Therefore, the lab should budget $12.125 per 100 rats. For 100 rats per week for 52 weeks, the total budget would be:
Total budget = $12.125 x 100 rats x 52 weeks = $63,350
Rounding to the nearest dollar yields the answer of B) $637.

Question 12

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 number of tickets written per day based on the Poisson distribution. If we sampled all days, we would expect the average number of tickets written to be 6.5 tickets per day.

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 hypergeometric distribution should be used in this case as it deals with sampling without replacement from a finite population. The analyst is selecting 3 companies from a group of 60, where the population size is fixed and known, and the success of one company affects the probability of success for the others. Therefore, the hypergeometric distribution is the appropriate choice for analyzing the probability of selecting the 3 companies that will eventually become highly successful.

Question 14

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 events occurring in a fixed interval of time or space, as long as the rate at which events occur is constant and events are independent of each other. Therefore, it is reasonable to use a Poisson distribution to model the number of customers arriving at a store in a given time period.

Question 15

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 mean cost of $13 per week represents the expected or average cost for all weekly rat purchases.

Question 16

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

Accepted Answer

The answer of If π remains constant in a binomial...

Question 17

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 answer of Suppose that the number of airplanes arriving...

Question 18

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 parameters 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) = 0.7361, which matches the given probability. Therefore, the answer is A, true.

Question 19

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, including infinity, with positive probability. Therefore, there is no maximum value that a Poisson random variable can have.

Question 20

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 would typically follow a Poisson distribution, not a binomial distribution. A binomial distribution involves a fixed number of trials with only two possible outcomes (success or failure) for each trial, while a Poisson distribution involves the number of occurrences of an event in a fixed time interval or space.

Quiz 5: Discrete Probability Distributions