Question 1

The standard deviation of a binomial distribution is _____.
A) E(x) = pn(1 − n)
B) E(x) = np(1 − p)
C) E(x) = np
D) None of the answers is correct.

Accepted Answer

The standard deviation of a binomial distribution is given by the square root of np(1-p), which is not correctly represented by any of the options provided.

Question 2

The binomial probability distribution is used with _____.
A) a continuous random variable
B) a discrete random variable
C) any distribution, as long as it is not normal
D) All of the answers are correct.

Accepted Answer

The binomial probability distribution is used with a discrete random variable. It models the outcomes of a fixed number of independent trials, each of which can have only two possible outcomes (success or failure).

Question 3

When using Excel's BINOM.DIST function, one should choose TRUE for the fourth input if _____.
A) a probability is desired
B) a cumulative probability is desired
C) the expected value is desired
D) the correct answer is desired

Accepted Answer

When using Excel's BINOM.DIST function, one should choose TRUE for the fourth input if a cumulative probability is desired. Choosing TRUE will give the cumulative probability up to and including the given value of x, while choosing FALSE will give the probability of getting exactly x successes.

Question 4

The standard deviation is the _____.
A) variance squared
B) square root of the sum of the deviations from the mean
C) same as the expected value
D) positive square root of the variance

Accepted Answer

The standard deviation is defined as the positive square root of the variance, which measures the spread of a set of data around the mean. Option A is incorrect because the variance is the average of the squared deviations, and taking the square root of that would not give us the standard deviation. Option B is only a part of the formula to calculate the standard deviation, but it is not the definition itself. Option C is incorrect because the expected value refers to the mean, but it is not the same as the standard deviation.

Question 5

Which of the following is NOT a property of a binomial experiment?
A) The experiment consists of a sequence of n identical trials.
B) Each outcome can be referred to as a success or a failure.
C) The probabilities of the two outcomes can change from one trial to the next.
D) The trials are independent.

Accepted Answer

In a binomial experiment, the probabilities of the two outcomes (success or failure) remain constant across all trials. They do not change from one trial to the next.

Question 6

Excel's BINOM.DIST function has how many inputs?
A) 2
B) 3
C) 4
D) 5

Accepted Answer

The BINOM.DIST function in Excel has 4 inputs: x (the number of successes), n (the number of trials), p (the probability of success), and cumulative (a logical value that determines whether to return the cumulative distribution function or the probability mass function). Therefore, the best choice is C, which indicates that there are 4 inputs.

Question 7

In a binomial experiment, the probability of success is .06. What is the probability of two successes in seven trials?
A) .0036
B) .06
C) .0554
D) .28

Accepted Answer

Using the binomial probability formula, we can find the probability of exactly 2 successes in 7 trials as follows:

P(2 successes) = (7 choose 2) * (0.06)^2 * (0.94)^5
P(2 successes) = 21 * 0.0036 * 0.734
P(2 successes) = 0.0554

Therefore, the probability of two successes in seven trials is 0.0554, which corresponds to choice C.

Question 8

Assume that you have a binomial experiment with p = 0.5 and a sample size of 100. The expected value of this distribution is _____.
A) 0.50
B) 0.30
C) 50
D) Not enough information is given to answer this question.

Accepted Answer

The formula for the expected value of a binomial distribution is: E(X) = np, where n is the sample size and p is the probability of success. Therefore, for this experiment, the expected value is: E(X) = 100(0.5) = 50. So, the answer is C.

Question 9

x is a random variable with the probability function: f(x) = x/6 for x = 1, 2, or 3. The expected value of x is _____.
A) 0.333
B) 0.500
C) 2.000
D) 2.333

Accepted Answer

The answer of x is a random variable with the...

Question 10

A probability distribution showing the probability of x successes in n trials, where the probability of success does not change from trial to trial, is termed a _____.
A) uniform probability distribution
B) binomial probability distribution
C) hypergeometric probability distribution
D) normal probability distribution

Accepted Answer

A binomial probability distribution is used to calculate the probability of a specific number of successes (x) in a fixed number of trials (n) with a constant probability of success on each trial. This fits the scenario described in the question. The other options (uniform, hypergeometric, and normal distributions) do not fit this scenario.

Question 11

Which of the following is NOT a characteristic of an experiment where the binomial probability distribution is applicable?
A) The experiment has a sequence of n identical trials.
B) Exactly two outcomes are possible on each trial.
C) The trials are dependent.
D) The probabilities of the outcomes do not change from one trial to another.

Accepted Answer

The binomial probability distribution is only applicable in experiments with independent trials, so choice C, which states that the trials are dependent, is not a characteristic of an experiment where the binomial probability distribution is applicable.

Question 12

Excel's BINOM.DIST function can be used to compute _____.
A) bin width for histograms
B) binomial probabilities
C) cumulative binomial probabilities
D) both binomial probabilities and cumulative binomial probabilities

Accepted Answer

BINOM.DIST function in Excel can be used to compute both binomial probabilities and cumulative binomial probabilities. Specifically, it returns the individual probability of a certain number of successes in a fixed number of trials, as well as the cumulative probability of that many successes or fewer.

Question 13

The variance for the binomial probability distribution is _____.
A) Var(x) = p(1 − p)
B) Var(x) = np
C) Var(x) = n(1 − p)
D) Var(x) = np(1 − p)

Accepted Answer

The formula for the variance of a binomial probability distribution is Var(x) = np(1 − p).

Question 14

If you are conducting an experiment where the probability of a success is .02 and you are interested in the probability of two successes in 15 trials, the correct probability function to use is the _____.
A) standard normal probability density function
B) normal probability density function
C) Poisson probability function
D) binomial probability function

Accepted Answer

The situation described is a binomial probability problem, where there are a fixed number of trials (15), each trial has only two possible outcomes (success or failure), the probability of success is constant (.02), and the trials are independent. Therefore, the correct probability function to use is the binomial probability function.

Question 15

In a binomial experiment, the _____.
A) probability of success does not change from trial to trial
B) probability of success does change from trial to trial
C) probability of success could change from trial to trial, depending on the situation under consideration
D) All of the answers are correct.

Accepted Answer

A binomial experiment has a fixed number of trials and two possible outcomes: success or failure. The probability of success is constant for each trial.

Question 16

The expected value for a binomial probability distribution is _____.
A) E(x) = pn(1 − n)
B) E(x) = p(1 − p)
C) E(x) = np
D) E(x) = np(1 − p)

Accepted Answer

The expected value for a binomial probability distribution is given by E(x) = np, where n is the number of trials and p is the probability of success for each trial. Option A is incorrect as it has an incorrect formula. Option B is the variance formula. Option D is incorrect as it includes (1-p) which is not required for the expected value. Therefore, the correct answer is C.

Question 17

Four percent of the customers of a mortgage company default on their payments. A sample of five customers is selected. What is the probability that exactly two customers in the sample will default on their payments?
A) .2592
B) .0142
C) .9588
D) .7408

Accepted Answer

The problem can be solved using the binomial distribution formula, with n = 5 (sample size), p = 0.04 (probability of default), and x = 2 (number of defaults):

P(x=2) = (5 choose 2)*(0.04)^2*(1-0.04)^3 = 0.0142

Therefore, the probability that exactly two customers in the sample will default on their payments is 0.0142, which corresponds to choice (B).

Question 18

Which of the following is a characteristic of a binomial experiment?
A) At least two outcomes are possible.
B) The probability of success changes from trial to trial.
C) The trials are independent.
D) All of the answers are correct.

Accepted Answer

A binomial experiment has only two possible outcomes (success and failure), the probability of success remains constant throughout the experiment, each trial is independent, and the experiment consists of a fixed number of trials. Therefore, the correct characteristic of a binomial experiment is the independence of the trials (C).

Question 19

A production process produces 2% defective parts. A sample of five parts from the production process is selected. What is the probability that the sample contains exactly two defective parts?
A) .0004
B) .0038
C) .10
D) .02

Accepted Answer

We can use the binomial probability formula, which is: P(X = k) = nCk * p^k * (1-p)^(n-k), where n is the sample size, k is the number of successes (i.e. defective parts), p is the probability of success (i.e. 0.02), and nCk is the combination formula (i.e. number of ways to choose k items from a pool of n items). 
Substituting the values, we get: 
P(X = 2) = 5C2 * 0.02^2 * (1-0.02)^(5-2) = 0.0038
Therefore, the answer is B.

Question 20

The number of electrical outages in a city varies from day to day. Assume that the number of electrical outages (x) in the city has the following probability distribution. $$\begin{array} { c c } 
x & f ( x ) \
\hline 0 & 0.80 \
1 & 0.15 \
2 & 0.04 \
3 & 0.01
\end{array}$$ The mean and the standard deviation for the number of electrical outages (respectively) are _____.
A) 2.6 and 5.77
B) 0.26 and .577
C) 3 and .01
D) 0 and .8

Accepted Answer

The mean (μ) of a probability distribution is calculated as $\sum [x \cdot f(x)]$, and the standard deviation (σ) is the square root of the variance, which is $\sum [(x - \mu)^2 \cdot f(x)]$. For this distribution, $\mu = (0 \cdot 0.80) + (1 \cdot 0.15) + (2 \cdot 0.04) + (3 \cdot 0.01) = 0.26$. The variance is $(0 - 0.26)^2 \cdot 0.80 + (1 - 0.26)^2 \cdot 0.15 + (2 - 0.26)^2 \cdot 0.04 + (3 - 0.26)^2 \cdot 0.01 = 0.3322$, and the standard deviation is the square root of 0.3322, which is approximately 0.577. Therefore, the mean is 0.26 and the standard deviation is approximately 0.577.

Quiz 5: Discrete Probability Distributions

The standard deviation of a binomial distribution is _____.

The binomial probability distribution is used with _____.

When using Excel's BINOM.DIST function, one should choose TRUE for the fourth input if _____.

The standard deviation is the _____.

Which of the following is NOT a property of a binomial experiment?

Excel's BINOM.DIST function has how many inputs?

In a binomial experiment, the probability of success is .06. What is the probability of two successes in seven trials?

Assume that you have a binomial experiment with p = 0.5 and a sample size of 100. The expected value of this distribution is _____.

x is a random variable with the probability function: f(x) = x/6 for x = 1, 2, or 3. The expected value of x is _____.

A probability distribution showing the probability of x successes in n trials, where the probability of success does not change from trial to trial, is termed a _____.

Which of the following is NOT a characteristic of an experiment where the binomial probability distribution is applicable?

Excel's BINOM.DIST function can be used to compute _____.

The variance for the binomial probability distribution is _____.

If you are conducting an experiment where the probability of a success is .02 and you are interested in the probability of two successes in 15 trials, the correct probability function to use is the _____.

In a binomial experiment, the _____.

The expected value for a binomial probability distribution is _____.

Four percent of the customers of a mortgage company default on their payments. A sample of five customers is selected. What is the probability that exactly two customers in the sample will default on their payments?

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

A production process produces 2% defective parts. A sample of five parts from the production process is selected. What is the probability that the sample contains exactly two defective parts?