Question 1

If n = 20 and p = .4,then the mean of the binomial distribution is

Accepted Answer

The answer of If n = 20 and p =...

Question 2

Which of the following statements about the binomial distribution is not correct?

Accepted Answer

The binomial distribution is a discrete probability distribution with a finite number of possible outcomes. Therefore, the random variable of interest is not continuous.

Question 3

A fair die is rolled 36 times.What is the standard deviation of the even number (2,4,or 6)outcomes?

Accepted Answer

The probability of rolling an even number on a fair die is 1/2. Therefore, the expected value of the number of even outcomes in 36 rolls is (1/2) * 36 = 18.

To calculate the standard deviation of the even outcomes, we need to use the formula:

SD = sqrt [ Σ(x - μ)^2 / N ]

where: 
- x is the number of even outcomes (2, 4, or 6) 
- μ is the mean of the even outcomes, which is 18 
- N is the number of rolls, which is 36

Let's fill in the formula:

SD = sqrt [ ( (2-18)^2 + (4-18)^2 + (6-18)^2 ) / 36 ]

= sqrt [ ( 16 + 196 + 256 ) / 36 ]

= sqrt [ 24 ]

= 4.899

Rounding this to the nearest whole number gives us a standard deviation of 5. Therefore, the answer is D.

Question 4

A fair die is rolled 10 times.What is the probability that an even number (2,4,or 6)will occur between 2 and 4 times?

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The answer of A fair die is rolled 10 times.What...

Question 5

The Securities and Exchange Commission has determined that the number of companies listed in NYSE declaring bankruptcy is approximately a Poisson distribution with a mean of 2.6 per month. Find the probability that no more than one bankruptcy occurs next month.

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The answer of The Securities and Exchange Commission has determined...

Question 6

The manager of the local grocery store has determined that,on the average,4 customers use the service desk every half-hour.Assume that the number of customers using the service desk has a Poisson distribution. What is the probability that during a randomly selected half-hour period no more than 2 customers use the service desk?

Accepted Answer

The probability of no more than 2 customers using the service desk can be calculated using the Poisson probability formula. With an average rate (λ) of 4 customers per half-hour, the probability of 0, 1, or 2 customers is calculated and summed: P(X ≤ 2) = P(X=0) + P(X=1) + P(X=2). This results in approximately 0.2381.

Question 7

Whenever p = .5,the binomial distribution will _________ be symmetric.

Accepted Answer

When p=.5, the probability of success and failure is equal. This results in a symmetric distribution where the mean and median are equal, and values on one side of the mean are mirrored on the other side. Therefore, the binomial distribution will always be symmetric when p=.5.

Question 8

The Securities and Exchange Commission has determined that the number of companies listed in NYSE declaring bankruptcy is approximately a Poisson distribution with a mean of 2.6 per month. Find the probability that more than 1 bankruptcy occur next month.

Accepted Answer

The probability of more than 1 bankruptcy is 1 minus the probability of 0 or 1 bankruptcy. Using the Poisson formula, P(X=k) = (e^(-λ) * λ^k) / k!, where λ = 2.6. Calculate P(X=0) and P(X=1), then subtract their sum from 1 to find P(X>1).

Question 9

The Securities and Exchange Commission has determined that the number of companies listed in NYSE declaring bankruptcy is approximately a Poisson distribution with a mean of 2.6 per month. Find the probability that exactly 4 bankruptcies occur next month.

Accepted Answer

The probability of observing exactly 4 bankruptcies next month can be calculated as:
P(X=4) = (e^-2.6)*(2.6^4)/(4!) = 0.1414
where X is the number of bankruptcies in a month, e is the base of the natural logarithm, and ! denotes the factorial function. Therefore, the correct choice is B).

Question 10

If n = 15 and p = .4,then the standard deviation of the binomial distribution is

Accepted Answer

The standard deviation of a binomial distribution is calculated using the formula √(n * p * (1-p)). Substituting n = 15 and p = 0.4, we get √(15 * 0.4 * 0.6) = √3.6 ≈ 1.8974.

Question 11

In the most recent election,19% of all eligible college students voted.If a random sample of 20 students were surveyed: Find the probability that exactly half voted in the election.

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The answer of In the most recent election,19% of all...

Question 12

In the most recent election,19% of all eligible college students voted.If a random sample of 20 students were surveyed: Find the probability that none of the students voted.

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The answer of In the most recent election,19% of all...

Question 13

The probability that a given computer chip will fail is 0.02.Find the probability that of 5 delivered chips,exactly 2 will fail.

Accepted Answer

The answer of The probability that a given computer chip...

Question 14

Two characteristics/assumptions of the Poisson distribution include:

Accepted Answer

The two characteristics/assumptions of the Poisson distribution are that the event occurring in one interval is independent of the event occurring in any other non-overlapping interval, and the random variable of interest is discrete. The probability of success (i.e. the event occurring) does not need to remain constant from trial to trial, as this is a property of the binomial distribution. The random variable of interest in a Poisson distribution is also not continuous, as it takes discrete values (i.e. the number of events that occur in a given interval). The event occurring in one interval is not dependent on the event occurring in any other non-overlapping interval, as this would violate the independence assumption that is necessary for a Poisson distribution.

Question 15

Of all individual tax returns,37% include errors made by the taxpayer.If IRS examiners are assigned randomly selected returns in batches of 12,find the mean and standard deviation for the number of erroneous returns per batch.

Accepted Answer

= 7.56,= 2.80

Question 16

A fair die is rolled 10 times.What is the probability that an odd number (1,3,or 5)will occur less than 3 times?

Accepted Answer

The probability of rolling an odd number on a fair die is 3/6 or 1/2. The probability of rolling an odd number less than 3 times in 10 rolls can be calculated by finding the probability of rolling 0, 1, or 2 odd numbers.

P(0 odds) = (1/2)^10 = 1/1024
P(1 odd) = 10*(1/2)^10 = 10/1024
P(2 odds) = 45*(1/2)^10 = 45/1024

Therefore, P(less than 3 odds) = P(0 odds) + P(1 odd) + P(2 odds) = 1/1024 + 10/1024 + 45/1024 = 56/1024 = 0.0547.

Option A) is the closest choice to this probability.

Question 17

A fair die is rolled 10 times.What is the average number of even number outcomes?

Accepted Answer

The probability of rolling an even number is 1/2, and the probability of rolling an odd number is also 1/2. Therefore, on average, we would expect 5 even number outcomes out of 10 rolls, since (10 rolls * 1/2 = 5 even number outcomes). Therefore, C is the best choice.

Question 18

The equation for the variance of the binomial distribution is given by:

Accepted Answer

The equation for the variance of the binomial distribution is np(1-p), where n is the number of trials and p is the probability of success on each trial. Option A appears to be an incorrect formula. Option B is only the expected value or mean of the distribution. Option D is not a valid choice as it is not one of the given options. Therefore, option C is the best choice as it correctly represents the formula for variance of the binomial distribution.

Question 19

The manager of the local grocery store has determined that,on the average,4 customers use the service desk every half-hour.Assume that the number of customers using the service desk has a Poisson distribution. What is the probability that during a randomly selected half-hour period exactly 2 customers use the service desk?

Accepted Answer

The probability of exactly k events in a Poisson distribution is given by the formula P(X=k) = (λ^k * e^(-λ)) / k!, where λ is the average rate (4 in this case) and k is the number of events (2 in this case). Plugging in the values, P(X=2) = (4^2 * e^(-4)) / 2! = 0.1465.

Question 20

According to a survey of adults,64% have money in a regular savings account.If we plan on surveying 50 randomly selected adults,find the mean number of adults who would have regular savings accounts.

Accepted Answer

The mean number of adults with regular savings accounts is calculated by multiplying the total number of surveyed adults (50) by the probability (64% or 0.64), which equals 32.

Quiz 5: Discrete Random Variables