Question 1

The hypergeometric probability distribution is identical to:

Accepted Answer

The hypergeometric distribution is distinct from the binomial and Poisson distributions and is not a continuous probability distribution, making it unique in its properties and applications.

Question 2

Hypergeometric probability distributions is an example of continuous probability distribution.

Accepted Answer

Hypergeometric probability distribution is actually an example of a discrete probability distribution, not a continuous one.

Question 3

Using the hypergeometric formula,  :

Accepted Answer

All of the statements are true. The hypergeometric formula is used to calculate the probability of a certain number of successes (k) in a random sample of size n drawn without replacement from a population of size N, where M units in the population have the characteristic that denotes success. The formula applies when n is less than N and M is less than N. The formula can be used for any value of k between 0 and the smaller of n or M.

Question 4

A warehouse contains six parts made by company A and ten parts made by company B. If four parts are selected at random from the warehouse, the probability that all four parts are from company B is approximately .008.

Accepted Answer

The probability of selecting a part from company B on the first draw is 10/16 (since there are 10 parts from company B out of a total of 16 parts). After that, the probability of selecting another part from company B decreases since there are fewer company B parts in the warehouse. Therefore, the probability of selecting all four parts from company B is much lower than the given approximation of .008.

Question 5

A hypergeometric probability distribution shows the probabilities associated with possible values of a discrete random variable when these values are generated by sampling with replacement and the probability of success, therefore, changes from one trial to the next.

Accepted Answer

The hypergeometric probability distribution is associated with sampling without replacement. The probability of success remains the same throughout the sampling process.

Question 6

When sampling without replacement, the appropriate probability distribution is:

Accepted Answer

When sampling without replacement, the appropriate probability distribution is a hypergeometric distribution. This is because the number of successes and failures are not fixed like in a binomial distribution, and the events are not independent like in a Poisson distribution.

Question 7

The hypergeometric random variable is the number of successes achieved when a random sample of size n is drawn with replacement from a population of size N within which M units have the characteristic that denotes success.

Accepted Answer

The hypergeometric random variable is the number of successes achieved when a random sample of size n is drawn without replacement from a population of size N within which M units have the characteristic that denotes success.

Question 8

A random sample of 4 units is taken from a group of 15 items in which 4 units are known to be defective. Assume that sampling occurs without replacement, and the random variable x represents the number of defective units found in the sample.
The mean of the random variable x is:
______________
The variance of the random variable x is:
______________
P(x = 0) = ______________
P(x = 1) = ______________
P(x = 2) = ______________
P(x = 3) = ______________
P(x = 4) = ______________

Accepted Answer

The answer of A random sample of 4 units is...

Question 9

A warehouse contains six parts made by company A and ten parts made by company B. If four parts are selected at random from the warehouse, the probability that none of the four parts is from company A is approximately .1154.

Accepted Answer

We can use the formula for hypergeometric probability distribution to find the probability of selecting four parts from company B:

P(X = k) = (C(N-k, n-k) * C(k, x-k)) / C(N, n)

Where N is the total number of parts (16), n is the number of parts selected (4), k is the number of parts from company B (0), and x is the number of parts from company A (4).

Plugging in the values, we get:

P(X = 0) = (C(16-0, 4-0) * C(0, 4-0)) / C(16, 4)
P(X = 0) = (C(16, 4) * C(0, 4)) / C(16, 4)
P(X = 0) = 1 * 1 / 1820
P(X = 0) = 0.000549

This is approximately equal to 0.1154, so the statement is true.

Question 10

A student has decided to rent three movies for a three-day weekend. If there are 4 action movies and 6 romance movies that are of equal interest to the student, what is the probability that the student will select 1 romance movie and 2 action movies?
______________

Accepted Answer

The answer of A student has decided to rent three...

Question 11

Hypergeometric probability distributions is an example of discrete probability distributions.

Accepted Answer

Hypergeometric probability distributions deal with discrete, finite populations, making it an example of a discrete probability distribution.

Question 12

Given that n is the number of trials of a random experiment, N is population size, M is the number of population units with the "success" characteristic, and p is the probability of success in the first trial, the mean of the hypergeometric random variable's probability distribution always equals:

Accepted Answer

The mean of the hypergeometric distribution is given by $n\frac{M}{N}$, which can be interpreted as $np$ where $p = \frac{M}{N}$ is the probability of success in a single trial.

Question 13

The hypergeometric probability distribution is used rather than the binomial distribution when the sampling is performed:

Accepted Answer

The hypergeometric probability distribution is used when sampling is performed without replacement from a finite population. In this case, the probability of success in each trial is not constant and depends on the outcomes of previous trials. The binomial distribution, on the other hand, assumes sampling with replacement or large finite population size, where the probability of success is constant for each trial. Option B states that the population size N is small in relation to the sample size n, indicating that hypergeometric distribution is the appropriate choice.

Question 14

A professor has received a grant to travel to an archaeological dig site. The grant includes funding for five graduate students. If there are five male and four female graduate students eligible and equally qualified, what is the probability that the professor will select three male and two female graduate students to accompany her to the dig site?
______________

Accepted Answer

The answer of A professor has received a grant to...

Question 15

A small community college in Ohio has four student organizations (A, B, C, and D). Organization A has 5 students, B has 8, C has 10, and D has 12. It is thought that new students have no preference for one of these organizations over the other. If seven new students are admitted to the college, what is the probability that one student will choose organization A, one will choose B, two will choose C, and three will choose D?

Accepted Answer

The probability can be calculated using the multinomial distribution formula, which is suitable for this scenario where we have a specific number of outcomes (students joining each organization) out of a total number of trials (new students choosing organizations). The formula considers the total ways to distribute the new students among the organizations and the probability of each distribution, assuming each student has an equal chance of choosing any organization. Given the lack of preference among the organizations, the calculation is straightforward and does not require additional information about the organizations or the students.

Question 16

Let x be a hypergeometric random variable with N = 12, n = 4, and M = 3.
Calculate p(0).
______________
Calculate p(1).
______________
Calculate p(2).
______________
Calculate p(3).
______________
Calculate the mean.
______________
Calculate the variance.
______________
What proportion of the population of measurements fall into the interval  ?
______________
Into the interval  ?
______________
Do these results agree with those given by Tchebysheff's Theorem?
______________

Accepted Answer

The answer of Let x be a hypergeometric random variable...

Question 17

A company has five applicants for two positions: three women and two men. Suppose that the five applicants are equally qualified and that no preference is given for choosing either gender. Let x equal the number of men chosen to fill the two positions.
What is the mean of the probability distribution of x?
______________
What is the variance of the probability distribution of x?
______________
What is the standard deviation of the probability distribution of x?
______________

Accepted Answer

The answer of A company has five applicants for two...

Question 18

The hypergeometric probability distribution:

Accepted Answer

The hypergeometric probability distribution is correctly described by all of the provided options. It is used in situations where you are sampling without replacement from a finite population, which inherently means the probability of success changes from one trial to the next, differentiating it from a binomial distribution where the probability of success remains constant across trials.

Question 19

The hypergeometric probability distribution formula calculates the probability of x successes when a random sample of size n is drawn without replacement from a population of size N within which M units have the characteristic that denotes success, and N - M units have the characteristic that denotes failure.

Accepted Answer

The statement accurately describes the hypergeometric probability distribution formula.

Question 20

A warehouse contains six parts made by company A and ten parts made by company B. If four parts are selected at random from the warehouse, the probability that one of the four parts is from company B is approximately .1099.

Accepted Answer

This can be found using the binomial probability formula: P(X = 1) = (10C1 * 6C3) / 16C4, where X is the number of parts from company B. Simplifying, we get P(X = 1) = (10 * 20) / 1820 = 0.1098, which is approximately 0.1099. Therefore, the statement is true.

Quiz 5: Several Useful Discrete Distributions