Question 1

Which statement concerning the binomial distribution is correct?

Accepted Answer

The binomial distribution's probability density function covers all integer values of X from 0 to n. This is because it represents the probability of getting x successes in n trials, where each trial has a fixed probability of success (π). Therefore, the possible values of X range from 0 (when there are no successes) to n (when all trials are successes).

Question 2

For which binomial distribution would a Poisson approximation be unacceptable?

Accepted Answer

The Poisson approximation is generally acceptable when the sample size (n) is large and the probability of success (π) is small. However, when n is very large and π is relatively large as well, the Poisson approximation may not be accurate. In this case, option C has a sample size of 200 and a probability of success of 0.10, which is a relatively large proportion. Therefore, a Poisson approximation would not be acceptable for this binomial distribution.

Question 3

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?

Accepted Answer

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

Question 4

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

Accepted Answer

The coefficient of variation (CV) is defined as the ratio of the standard deviation to the mean, expressed as a percentage. For a Poisson distribution with parameter λ, the mean and standard deviation are both equal to λ. Therefore, the CV can be calculated as:

CV = (standard deviation / mean) * 100% = (λ / λ) * 100% = 100%

In this case, λ = 5, so the CV is 100%. However, since the choices given are all less than 100%, we need to convert the CV to a percentage relative to the mean:

CV% = 100% / λ * 100% = 20%

Therefore, the answer is (C), 44.7 percent, which is closest to 20%.

Question 5

If X is a discrete uniform random variable ranging from 0 to 12,find P(X ≥ 10).

Accepted Answer

Since X is a uniform random variable ranging from 0 to 12, each value in this range is equally likely to occur. Therefore, the probability of X being greater than or equal to 10 is equal to the number of values in the range that are greater than or equal to 10 divided by the total number of values in the range.

There are three values in the range 0 to 12 that are greater than or equal to 10: 10, 11, and 12. 
Total number of values in the range = 13 (0,1,2,3....12)

Therefore, P(X ≥ 10) = 3/13 = 0.2308.

Hence, the answer is choice C.

Question 6

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?

Accepted Answer

The answer of On a Sunday in April,dog bite victims...

Question 7

If X is a discrete uniform random variable ranging from 12 to 24,what is its mean?

Accepted Answer

The formula for the mean of a discrete uniform random variable is (min value + max value) / 2. 
Therefore, the mean of X is (12 + 24) / 2 = 18.

Question 8

If X is a discrete uniform random variable ranging from one to eight,find P(X < 6).

Accepted Answer

The probability P(X < 6) is the number of favorable outcomes (1, 2, 3, 4, 5) divided by the total number of outcomes (1 to 8), which is 5/8 = 0.625.

Question 9

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?

Accepted Answer

The given situation follows a Poisson distribution with λ = 4.7 (average number of fraudulent returns discovered per day). Using the Poisson probability formula, we can find the probability of discovering fewer than two fraudulent returns:
P(X < 2) = e^(-λ) * Σ(λ^k / k!), k=0 to 1
P(X < 2) = e^(-4.7) * [(4.7^0 / 0!) + (4.7^1 / 1!)]
P(X < 2) = 0.0518 (rounded to 4 decimal places)
Therefore, the best choice is A.

Question 10

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?

Accepted Answer

The number of cars that arrive in a given amount of time follows a Poisson distribution with a mean of λ = (rate of arrivals)×(time interval).
In this case, λ = 4 cars/minute × 2 minutes = 8 cars.
The probability of exactly eight cars arriving in the next two minutes can be calculated using the Poisson distribution formula:
P(X=8) = e^(-λ) × (λ^8) / 8!
= e^(-8) × (8^8) / 8!
= 0.1396 (rounded to four decimal places)

Question 11

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?

Accepted Answer

The problem follows a Poisson distribution with λ = 1.6 (since 16 defects per 100 meters translates to 1.6 defects per 10 meters). The probability of finding exactly 2 defects is calculated using the Poisson probability formula: P(X=k) = (e^(-λ) * λ^k) / k!. Plugging in the values, P(X=2) = (e^(-1.6) * 1.6^2) / 2! = 0.2584.

Question 12

The true proportion of accounts receivable with some kind of error is 0.02 for Venal Enterprises.If an auditor randomly samples 50 accounts receivable,which is the Excel probability that fewer than two will contain errors?

Accepted Answer

The correct function to find the probability that fewer than two accounts contain errors is BINOM.DIST(1,50,0.02,1), which calculates the cumulative probability for 0 and 1 errors.

Question 13

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

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

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

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 15

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

Accepted Answer

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

Question 16

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

Accepted Answer

The domain of X in a Poisson probability distribution is discrete and can only include nonnegative integer values. Therefore, choice C is the best choice.

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?

Accepted Answer

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

Question 18

For which binomial distribution would a Poisson approximation be acceptable?

Accepted Answer

A Poisson approximation for a binomial distribution is acceptable when n is large and π is small. In this case, n = 40 and π = 0.03, so a Poisson approximation would be acceptable. The other choices have either a small n or a relatively large π, making a Poisson approximation less accurate.

Question 19

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.

Accepted Answer

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

Question 20

If X is a discrete uniform random variable ranging from one to eight,what is its mean?

Accepted Answer

The formula for the mean of a discrete uniform random variable is (a + b)/2, where a is the minimum value and b is the maximum value. In this case, a = 1 and b = 8. Therefore, the mean of X is (1 + 8)/2 = 4.5, which is option B.

Quiz 6: Discrete Probability Distributions