Question 1

The coefficient of variation for a Poisson distribution with λ = 5 is:
A) 35.2 percent.
B) 58.9 percent.
C) 44.7 percent.
D) 31.1 percent.

Accepted Answer

The formula for coefficient of variation is: 
CV = (standard deviation/mean) x 100%
For Poisson distribution, the variance is λ (which is also the mean).
Therefore, standard deviation = sqrt(λ) = sqrt(5)
CV = (sqrt(5)/5) x 100% = 44.7%

Question 2

For which binomial distribution would a Poisson approximation be acceptable?
A) n = 60,π = 0.08
B) n = 100,π = 0.15
C) n = 40,π = 0.03
D) n = 20,π = 0.20

Accepted Answer

A Poisson distribution can be used to approximate a binomial distribution when the sample size (n) is large and the success probability (π) is small, typically less than 0.1. Option C has a small success probability of 0.03 and a sample size of 40, making it the best choice for a Poisson approximation. The other options have larger success probabilities and/or smaller sample sizes, making them less suitable for a Poisson approximation.

Question 3

On average,an IRS auditor discovers 4.7 fraudulent income tax returns per day.On a randomly chosen day,what is the probability that she discovers fewer than two?
A) .0518
B) .0427
C) .1005
D) .1523

Accepted Answer

The scenario describes a Poisson distribution with a mean ($\lambda$) of 4.7 fraudulent returns per day. The probability of observing fewer than two fraudulent returns ($k < 2$) is the sum of the probabilities of observing 0 and 1 fraudulent returns. Using the Poisson probability formula $P(k; \lambda) = \frac{e^{-\lambda} \lambda^k}{k!}$, we calculate $P(0; 4.7) + P(1; 4.7)$. This calculation results in a value that matches choice A, indicating a misunderstanding or a mistake in the calculation process provided in the options.

Question 4

The probability that a rental car will be stolen is 0.0004.If 3500 cars are rented,what is the approximate Poisson probability that 2 or fewer will be stolen?
A) .3452
B) .2417
C) .5918
D) .8335

Accepted Answer

The Poisson probability formula is $P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$, where $\lambda$ is the average rate (mean) of occurrence and $k$ is the number of occurrences. Here, $\lambda = 3500 \times 0.0004 = 1.4$. To find the probability of 2 or fewer cars being stolen, we calculate the sum of probabilities for 0, 1, and 2 cars being stolen using the Poisson formula and sum them up. The calculation yields approximately 0.835, or 83.35% when converted to a percentage, which corresponds to choice D.

Question 5

Historically,2 percent of the stray dogs in Southfield are unlicensed.On a randomly chosen day,the Southfield city animal control officer picks up seven stray dogs.What is the probability that fewer than two will be unlicensed?
A) .8681
B) .9921
C) .3670
D) .0076

Accepted Answer

The answer of Historically,2 percent of the stray dogs in...

Question 6

The coefficient of variation for a Poisson distribution with λ = 4 is:
A) 35.2 percent.
B) 58.9 percent.
C) 50.0 percent.
D) 26.4 percent.

Accepted Answer

The answer of The coefficient of variation for a Poisson...

Question 7

On average,a major earthquake (Richter scale 6.0 or above)occurs three times a decade in a certain California county.Find the probability that at least one major earthquake will occur within the next decade.
A) .7408
B) .1992
C) .1494
D) .9502

Accepted Answer

The answer of On average,a major earthquake (Richter scale 6.0...

Question 8

If tubing averages 16 defects per 100 meters,what is the probability of finding exactly 2 defects in a randomly chosen 10-meter piece of tubing?
A) .8795
B) .2674
C) .3422
D) .2584

Accepted Answer

The answer of If tubing averages 16 defects per 100...

Question 9

For which binomial distribution would a Poisson approximation not be acceptable?
A) n = 35,π = 0.07
B) n = 95,π = 0.01
C) n = 80,π = 0.02
D) n = 50,π = 0.03

Accepted Answer

For a Poisson approximation to be acceptable, the expected value (λ = nπ) should be at least 5. In this case, λ = 35 x 0.07 = 2.45 which is less than 5. Therefore, a Poisson approximation would not be acceptable for this binomial distribution.

Question 10

The true proportion of accounts receivable with some kind of error is .02 for Venal Enterprises.If an auditor randomly samples 200 accounts receivable,what is the approximate Poisson probability that fewer than two will contain errors?
A) .1038
B) .0916
C) .1465
D) .0015

Accepted Answer

We can use the Poisson distribution with lambda = np = 200 * 0.02 = 4. The probability of fewer than two errors is P(X < 2) = P(X = 0) + P(X = 1) which is approximately 0.0916, using a Poisson calculator or the Poisson probability function in Excel. The answer is therefore choice B.

Question 11

Cars are arriving at a toll booth at a rate of four per minute.What is the probability that exactly eight cars will arrive in the next two minutes?
A) 0.0349
B) 0.1396
C) 0.9666
D) 0.0005

Accepted Answer

The number of cars arriving in two minutes follows a Poisson distribution with a mean of 8 (4 cars per minute x 2 minutes). Using the Poisson probability formula, P(X = 8) = (e^-8) (8^8) / 8! ≈ 0.1396. Therefore, the best choice is B.

Question 12

The domain of X in a Poisson probability distribution is discrete and can include:
A) any real X value.
B) any integer X value.
C) any nonnegative integer X value.
D) any X value except zero.

Accepted Answer

The domain of X in a Poisson probability distribution is discrete and can only include nonnegative integer values because the Poisson distribution models the number of occurrences of an event in a given time or space interval, which cannot be negative or fractional.

Question 13

Arrival of cars per minute at a toll booth may be characterized by the Poisson distribution if:
A) the arrivals are independent.
B) no more than one arrival can occur in a minute.
C) there is only one lane leading to the booth.
D) the mean arrival rate is at least 30.

Accepted Answer

The Poisson distribution is used to model the number of events occurring within a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.

Question 14

On a Sunday in April,dog bite victims arrive at Carver Memorial Hospital at a historical rate of 0.6 victim per day.On a given Sunday in April,what is the probability that exactly two dog bite victims will arrive?
A) .0875
B) .0902
C) .0988
D) .0919

Accepted Answer

The probability of exactly two dog bite victims arriving at Carver Memorial Hospital on a given Sunday in April, when the historical rate is 0.6 victim per day, can be calculated using the Poisson distribution formula: P(x; μ) = (e^(-μ) * μ^x) / x!, where μ is the average rate (0.6 in this case), x is the number of occurrences (2 victims), and e is the base of the natural logarithm (approximately 2.71828). Plugging in the values gives P(2; 0.6) = (e^(-0.6) * 0.6^2) / 2! = (0.5488 * 0.36) / 2 = 0.0988.

Question 15

Which statement concerning the binomial distribution is correct?
A) Its PDF covers all integer values of X from 0 to n.
B) Its PDF is the same as its CDF when π = .50.
C) Its CDF shows the probability of each value of X.
D) Its CDF is skewed right when π < .50.

Accepted Answer

The binomial distribution covers all integer values of X from 0 to n, where n is the number of trials and X is the number of successes. The PDF (probability density function) gives the probability of getting exactly X successes in n trials. The CDF (cumulative distribution function) shows the probability of getting up to and including X successes in n trials. The skewness of the distribution depends on the value of π, the probability of success for each trial, but not on whether the CDF or PDF is used.

Question 16

For which binomial distribution would a Poisson approximation be unacceptable?
A) n = 30,π = 0.02
B) n = 50,π = 0.03
C) n = 200,π = 0.10
D) n = 500,π = 0.01

Accepted Answer

The Poisson approximation to the binomial distribution is generally considered acceptable if n is large and π is small, with the product nπ (mean of the binomial distribution) not too large. In option C, n = 200 and π = 0.10, which does not meet the criteria for a small π as well as the product nπ = 20, which can be considered too large for an ideal Poisson approximation.

Question 17

On Saturday morning,calls arrive at TicketMaster at a rate of 108 calls per hour.What is the probability of fewer than three calls in a randomly chosen minute?
A) .1607
B) .8913
C) .2678
D) .7306

Accepted Answer

The answer of On Saturday morning,calls arrive at TicketMaster at...

Question 18

The probability that a ticket holder will miss a flight is .005.If 180 passengers take the flight,what is the approximate Poisson probability that at least 2 will miss the flight?
A) .9372
B) .0628
C) .1647
D) .2275

Accepted Answer

The answer of The probability that a ticket holder will...

Question 19

The probability that a customer will use a stolen credit card to make a purchase at a certain Target store is 0.003.If 400 purchases are made in a given day,what is the approximate Poisson probability that 4 or fewer will be with stolen cards?
A) .0053
B) .0076
C) .9923
D) .0555

Accepted Answer

The Poisson probability formula is used here, given by P(X=k) = (λ^k * e^(-λ)) / k!, where λ is the mean number of successes (λ = np), and k is the number of successes we're interested in. For this scenario, λ = 400 * 0.003 = 1.2. We're interested in the probability of 4 or fewer successes, which requires summing the probabilities for 0, 1, 2, 3, and 4 successes. This calculation is cumbersome by hand but is well approximated by Poisson probability tables or a calculator. The result is approximately 0.9923, indicating a high probability of 4 or fewer purchases with stolen cards.

Question 20

The probability that a certain daily flight's departure from ORD to LAX is delayed is .02.Over six months,this flight departs 180 times.What is the approximate Poisson probability that it will be delayed fewer than 2 times?
A) .4471
B) .3028
C) .1257
D) .1771

Accepted Answer

Using the Poisson distribution, where λ = mean number of delays in 180 flights = 180 x 0.02 = 3.6:

P(X < 2) = P(X=0) + P(X=1)
= (e^-λ * λ^0 / 0!) + (e^-λ * λ^1 / 1!)
= (e^-3.6 * 3.6^0 / 0!) + (e^-3.6 * 3.6^1 / 1!)
= (0.0273) + (0.0981)
= 0.1254

Therefore, the approximate Poisson probability that the flight will be delayed fewer than 2 times is 0.1257 (rounded to four decimal places). Answer choice C is the closest value to this answer.

Quiz 6: Discrete Probability Distributions

The coefficient of variation for a Poisson distribution with λ = 5 is:

For which binomial distribution would a Poisson approximation be acceptable?

On average,an IRS auditor discovers 4.7 fraudulent income tax returns per day.On a randomly chosen day,what is the probability that she discovers fewer than two?

The probability that a rental car will be stolen is 0.0004.If 3500 cars are rented,what is the approximate Poisson probability that 2 or fewer will be stolen?

Historically,2 percent of the stray dogs in Southfield are unlicensed.On a randomly chosen day,the Southfield city animal control officer picks up seven stray dogs.What is the probability that fewer than two will be unlicensed?

The coefficient of variation for a Poisson distribution with λ = 4 is:

On average,a major earthquake (Richter scale 6.0 or above) occurs three times a decade in a certain California county.Find the probability that at least one major earthquake will occur within the next decade.

If tubing averages 16 defects per 100 meters,what is the probability of finding exactly 2 defects in a randomly chosen 10-meter piece of tubing?

For which binomial distribution would a Poisson approximation not be acceptable?

The true proportion of accounts receivable with some kind of error is .02 for Venal Enterprises.If an auditor randomly samples 200 accounts receivable,what is the approximate Poisson probability that fewer than two will contain errors?

Cars are arriving at a toll booth at a rate of four per minute.What is the probability that exactly eight cars will arrive in the next two minutes?

The domain of X in a Poisson probability distribution is discrete and can include:

Arrival of cars per minute at a toll booth may be characterized by the Poisson distribution if:

On a Sunday in April,dog bite victims arrive at Carver Memorial Hospital at a historical rate of 0.6 victim per day.On a given Sunday in April,what is the probability that exactly two dog bite victims will arrive?

Which statement concerning the binomial distribution is correct?

For which binomial distribution would a Poisson approximation be unacceptable?

On Saturday morning,calls arrive at TicketMaster at a rate of 108 calls per hour.What is the probability of fewer than three calls in a randomly chosen minute?

The probability that a ticket holder will miss a flight is .005.If 180 passengers take the flight,what is the approximate Poisson probability that at least 2 will miss the flight?

The probability that a customer will use a stolen credit card to make a purchase at a certain Target store is 0.003.If 400 purchases are made in a given day,what is the approximate Poisson probability that 4 or fewer will be with stolen cards?

The probability that a certain daily flight's departure from ORD to LAX is delayed is .02.Over six months,this flight departs 180 times.What is the approximate Poisson probability that it will be delayed fewer than 2 times?