Question 1

The binomial distribution could be used to describe the spread of tennis balls when the players are serving.

Accepted Answer

The binomial distribution is used for discrete random variables, typically representing the number of successes in a fixed number of independent Bernoulli trials, where each trial has only two possible outcomes (success or failure). The spread of tennis balls when players are serving involves continuous outcomes (the locations where the balls land), which would be better described by a continuous probability distribution, not a binomial distribution.

Question 2

Which of the following best describes a binomial probability distribution?
A) It is a probability distribution that shows the probabilities associated with possible values of a discrete random variable that are generated by a binomial experiment.
B) It is a probability distribution that shows, for each of a random variable's possible values, the probability of the variable being less than or equal to that value.
C) It is a probability distribution that shows the probabilities associated with possible values of a discrete random variable when these values are generated by sampling without replacement and the probability of success, therefore, changes from one trial to the next.
D) It is a probability distribution that shows the probabilities associated with possible values of a discrete random variable when these values equal the number of occurrences of a specified event within a specified time or space.
E) It is a probability distribution that shows, for each of a random variable's possible values, the probability of the variable is positive.

Accepted Answer

A binomial probability distribution is a probability distribution that shows the probabilities associated with possible values of a discrete random variable that are generated by a binomial experiment. In a binomial experiment, there are only two possible outcomes (success or failure) for each trial, the trials are independent of one another, the probability of success remains constant for all trials, and the experiment consists of a fixed number of trials. The binomial distribution is used to model the number of successes in a fixed number of independent trials with a constant probability of success.

Question 3

For a binomial experiment with n trials, p is the probability of success, q is the probability of failure, and x is the number of successes in n trials. Which one of the following statements is correct?
A) p + q = 1
B) p(x) = 1 for x = 0, 1, . . ., n
C) P(x = 0) =
D) p + q = 1 and p(x) = 1 for x = 0, 1, . . ., n
E) all of these

Accepted Answer

The binomial distribution is a discrete probability distribution that describes the number of successes in a sequence of independent experiments, each of which has a constant probability of success. The probability of success is denoted by p, and the probability of failure is denoted by q. The total number of trials is denoted by n.The probability of obtaining x successes in n trials is given by the binomial probability mass function:```P(X = x) = (n! / x!(n - x)!) * p^x * q^(n - x)```where:* n is the number of trials* x is the number of successes* p is the probability of success* q is the probability of failureThe sum of the probabilities of all possible outcomes must equal 1. Therefore, we have:```∑[P(X = x)] = 1```This means that the sum of the probabilities of obtaining 0, 1, 2, ..., n successes must equal 1.The probability of obtaining 0 successes is given by:```P(X = 0) = (n! / 0!(n - 0)!) * p^0 * q^(n - 0) = q^n```Therefore, the probability of obtaining 0 successes is q^n.The probability of obtaining 1 success is given by:```P(X = 1) = (n! / 1!(n - 1)!) * p^1 * q^(n - 1) = n * p * q^(n - 1)```Therefore, the probability of obtaining 1 success is n * p * q^(n - 1).The probability of obtaining 2 successes is given by:```P(X = 2) = (n! / 2!(n - 2)!) * p^2 * q^(n - 2) = n * (n - 1) * p^2 * q^(n - 2)```Therefore, the probability of obtaining 2 successes is n * (n - 1) * p^2 * q^(n - 2).And so on.The probability of obtaining x successes in n trials is given by the binomial probability mass function. The sum of the probabilities of all possible outcomes must equal 1. The probability of obtaining 0 successes is q^n. The probability of obtaining 1 success is n * p * q^(n - 1). The probability of obtaining 2 successes is n * (n - 1) * p^2 * q^(n - 2). And so on.

Question 4

Given that n is the number of trials and p is the probability of success in any one trial of a random experiment, the expected value of a binomial random variable equals:
A) n
B) p
C) np
D) n/p
E) n/p(1 - p)

Accepted Answer

The expected value of a binomial random variable is given by the formula E(X) = np. This means that the expected value is equal to the number of trials multiplied by the probability of success in each trial. Therefore, the correct option is C.

Question 5

Which of the following best describes the binomial probability distribution?
A) If is a probability distribution that shows the probabilities associated with possible values of a discrete random variable when these values are generated by sampling without replacement and the probability of success, therefore, changes from one trial to the next.
B) It is a probability distribution that shows the probabilities associated with possible values of a discrete random variable when these values equal the number of occurrences of a specified event within a specified time or space.
C) It is a probability distribution that shows the probabilities associated with possible values of a discrete random variable that are generated by a binomial experiment.
D) It is a probability distribution that shows the probabilities associated with possible values of a continuous random variable that are generated by a binomial experiment.
E) None of these.

Accepted Answer

The binomial probability distribution is specifically associated with outcomes from a binomial experiment, which involves a fixed number of independent trials with only two possible outcomes (usually labeled success and failure) and a fixed probability of success on each trial. The distribution shows the probabilities associated with possible numbers of successes in a given number of trials, making it a discrete probability distribution.

Question 6

The binomial distribution is used to describe continuous random variables.

Accepted Answer

The binomial distribution is used to describe discrete random variables, not continuous random variables.

Question 7

Which of the following is a characteristic of a binomial problem?
A) There are n identical trials, and all trials are independent
B) Each trial has two possible outcomes which are traditionally labeled "failure" and "success" and the probability of success p is the same on each trial.
C) We are interested in x, the number of successes observed during the n trials.
D) All of these are characteristics of a binomial experiment.
E) None of these

Accepted Answer

A binomial problem has all of the characteristics listed in options A, B, and C.

Question 8

Which of the following statements regarding a binomial experiment is false, where n is the number of trials, and p is the probability of success in each trial?
A) The n trials are independent.
B) The standard deviation is np(1 - p).
C) The mean is np.
D) There are only two possible outcomes.
E) The n trials are independent and the standard deviation is np(1 - p).

Accepted Answer

The standard deviation of a binomial distribution is actually the square root of np(1 - p), not np(1 - p) as stated.

Question 9

The standard deviation of a binomial distribution for which n = 50 and p = 0.15 is:
A) 50.15
B) 7.082
C) 6.375
D) 2.525
E) 0.15

Accepted Answer

The standard deviation of a binomial distribution is calculated using the formula √(n * p * (1-p)). Substituting n = 50 and p = 0.15, we get √(50 * 0.15 * 0.85) = √(6.375) ≈ 2.525.

Question 10

The National Rifle Association chapter in a university town claims 60% of the students at the university favor gun control. A social scientist at the university conducts a survey of 20 randomly chosen students and finds 9 who favor gun control. Which of the following is a reasonable conclusion?
A) There is no reason to doubt the claim.
B) The survey results constitute a rare event.
C) The NRA's claim is incorrect.
D) The true percentage of students who favor gun control must be 45%.
E) None of these.

Accepted Answer

The sample size is small, and the observed proportion (9 out of 20) is not significantly different from the claimed 60%, so there is no strong evidence to doubt the NRA's claim.

Question 11

A telephone survey of American families is conducted to determine the number of children in the average American family. Past experience has shown that 30% of the families who are telephoned will refuse to respond to the survey. Which of the following statements is not a binomial random variable?
A) The number of families out of 50 who respond to the survey.
B) The number of families out of 50 who refuse to respond to the survey.
C) The number of children in a family which responds to the survey.
D) All of these are binomial random variables.
E) None of these is a binomial random variable.

Accepted Answer

The number of children in a family which responds to the survey is not a binomial random variable because it does not have a fixed number of trials with a binary outcome (success or failure). The other options all have a fixed number of trials (50 families surveyed) with a binary outcome (responding or not responding).

Question 12

The expected number of heads in 200 tosses of an unbiased coin is:
A) 100
B) 50
C) 75
D) 125
E) 200

Accepted Answer

An unbiased coin has a 50/50 chance of landing on either heads or tails on each toss. Therefore, the expected number of heads in 200 tosses would be 0.5 multiplied by 200, which equals 100.

Question 13

The number of defects in a random sample of 200 parts produced by a machine is binomially distributed with p = .03. Based on this information, the standard deviation of the number of defects in the sample is 5.82.

Accepted Answer

The standard deviation of a binomial distribution with parameters n and p is:

sigma = sqrt(np(1-p))

Using the given information, we can solve for n:

n = 200 
p = 0.03 
sigma = 5.82

5.82 = sqrt(200 * 0.03 * (1-0.03)) = 5.82

Therefore, the statement is true.

Question 14

The number of defects in a random sample of 200 parts produced by a machine is binomially distributed with p = .03. Based on this information, the expected number of defects in the sample is 6.

Accepted Answer

The expected value of a binomial distribution is given by np, where n is the sample size and p is the probability of success. In this case, n=200 and p=.03, so the expected value is 200*.03=6. Therefore, the statement is true.

Question 15

A life insurance salesperson makes 15 sales calls daily. The chance of making a sale on each call is 0.40. The probability that he will make at most 2 sales is less than 0.10.

Accepted Answer

This is a binomial distribution problem with n=15 and p=0.40. To find the probability of making at most 2 sales, we need to calculate P(X ≤ 2). Using a calculator or binomial table, we find that P(X ≤ 2) = 0.168, which is greater than 0.10. Therefore, the statement is false.

Question 16

The binomial distribution formula:
A) computes the number of permutations of x successes (and, therefore, n - x failures) that can be achieved in n trials of a random experiment that satisfies the conditions of the binomial experiment
B) computes the probability of x successes in n trials of a random experiment that satisfies the conditions of a binomial experiment
C) produces one of two possible outcomes that are conventionally called success and failure
D) computes 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
E) does all of these

Accepted Answer

The binomial distribution formula computes the probability of x successes in n trials of a random experiment that satisfies the conditions of a binomial experiment.

Question 17

The distribution of the number of phone calls to a doctor's office in a one-hour time period is likely to be described by a binomial distribution.

Accepted Answer

The binomial distribution applies to independent and identically distributed random variables, where there are a fixed number of trials and the outcomes are binary (success/failure). The number of phone calls to a doctor's office in a one-hour time period is not a binary outcome and may not be independent or identically distributed, making it unlikely to fit a binomial distribution. A more appropriate distribution could be Poisson or normal.

Question 18

The expected value, E(X), of a binomial probability distribution with n trials and probability p of success is:
A) n / p
B) np(1 - p)
C) np
D) np - 1
E) n(1 - p)

Accepted Answer

The expected value of a discrete probability distribution is the sum of the product of each outcome and its probability. For a binomial distribution with n trials, each trial has two possible outcomes (success or failure) with probability p and (1-p), respectively. Thus, the expected value can be calculated as E(X) = (0 x P(X=0)) + (1 x P(X=1)) + (2 x P(X=2)) + ... + (n x P(X=n)). Using the formula for binomial probabilities, we can write this as E(X) = 0((1-p)^n) + 1(np(1-p)^(n-1)) + 2((n choose 2)(p^2)(1-p)^(n-2)) + ... + n((n choose n)(p^n)). Simplifying this expression, we can factor out n from each term and simplify the binomial coefficients to get E(X) = np. Therefore, the correct answer is C) np.

Question 19

Which of the following is an example of a binomial experiment?
A) A shopping mall is interested in the income level of its customers and is taking a survey to gather information.
B) A business firm introducing a new product wants to know how many purchases its clients will make each year.
C) A sociologist is researching an area in an effort to determine the proportion of households with a male head of household.
D) A study is concerned with the average number of hours worked by high school students.
E) All of these.

Accepted Answer

A binomial experiment involves a fixed number of trials, each with two possible outcomes. Option C, determining the proportion of households with a male head of household, fits this description as each household either has a male head of household or does not, making it a binomial experiment.

Question 20

The binomial experiment requires that the successes and failure probabilities be constant from one trial to the next and also that these two probabilities be equal to each other.

Accepted Answer

The binomial experiment requires that the probability of success and failure remain constant across trials, but it does not require these probabilities to be equal to each other.

Quiz 5: Several Useful Discrete Distributions

The binomial distribution could be used to describe the spread of tennis balls when the players are serving.

Which of the following best describes a binomial probability distribution?

For a binomial experiment with n trials, p is the probability of success, q is the probability of failure, and x is the number of successes in n trials. Which one of the following statements is correct?

Given that n is the number of trials and p is the probability of success in any one trial of a random experiment, the expected value of a binomial random variable equals:

Which of the following best describes the binomial probability distribution?

The binomial distribution is used to describe continuous random variables.

Which of the following is a characteristic of a binomial problem?

Which of the following statements regarding a binomial experiment is false, where n is the number of trials, and p is the probability of success in each trial?

The standard deviation of a binomial distribution for which n = 50 and p = 0.15 is:

The National Rifle Association chapter in a university town claims 60% of the students at the university favor gun control. A social scientist at the university conducts a survey of 20 randomly chosen students and finds 9 who favor gun control. Which of the following is a reasonable conclusion?

A telephone survey of American families is conducted to determine the number of children in the average American family. Past experience has shown that 30% of the families who are telephoned will refuse to respond to the survey. Which of the following statements is not a binomial random variable?

The expected number of heads in 200 tosses of an unbiased coin is:

The number of defects in a random sample of 200 parts produced by a machine is binomially distributed with p = .03. Based on this information, the standard deviation of the number of defects in the sample is 5.82.

The number of defects in a random sample of 200 parts produced by a machine is binomially distributed with p = .03. Based on this information, the expected number of defects in the sample is 6.

A life insurance salesperson makes 15 sales calls daily. The chance of making a sale on each call is 0.40. The probability that he will make at most 2 sales is less than 0.10.

The binomial distribution formula:

The distribution of the number of phone calls to a doctor's office in a one-hour time period is likely to be described by a binomial distribution.

The expected value, E(X) , of a binomial probability distribution with n trials and probability p of success is:

Which of the following is an example of a binomial experiment?

The binomial experiment requires that the successes and failure probabilities be constant from one trial to the next and also that these two probabilities be equal to each other.