CH 5

b. a continuum of possible values.

A continuous probability distribution is characterized by a. counts. b. a continuum of possible values. c. a list of possible values. d. an array of individual values.

exponential distribution

A continuous probability distribution useful for measuring times between events, such as customer arrivals to a service facility; mean and standard deviation both equal the reciprocal of the parameter

d. is a set of possible values and a corresponding set of probabilities that sum to 1.

A discrete probability distribution a. can be estimated from long-run proportions. b. is a modeling tool that can be used to incorporate uncertainty into models. c. is the distribution of a single random variable. d. is a set of possible values and a corresponding set of probabilities that sum to 1.

b. random

A function that associates a numerical value with each possible outcome of an uncertain event is called a _____ variable. a. sample b. random c. population d. conditional

probability tree

A graphical representation of how events occur through time, useful for calculating probabilities of multiple events

standard deviation

A more natural measure of variability. denoted by σ or Stdev(X). It is the square root of the variance

a. True

A random variable X is normally distributed with a mean of 175 and a standard deviation of 50. Given that X = 150, its corresponding Z- score is -0.50. a. True b. False

a. True

A random variable X is standardized when each value of X has the mean of X subtracted from it, and the difference is divided by the standard deviation of X. a. True b. False

b. 68%

According to the empirical rule, how many observations lie within +/- 1 standard deviation from the mean? a. 50% b. 68% c. 95% d. Almost all

c. 95%

According to the empirical rule, how many observations lie within +/- 2 standard deviation from the mean? a. 50% b. 68% c. 95% d. Almost all

d. Almost all

According to the empirical rule, how many observations lie within +/- 3 standard deviation from the mean? a. 50% b. 68% c. 95% d. Almost all

b. Binomial distribution

An administrator at Lakeside Middle School is interested in examining the probability of a child being late to school. The child is categorized as either late or not late. What type of distribution should the school use to examine this issue? a. Exponential distribution b. Binomial distribution c. Normal distribution d. Poisson distribution

0,1

An event with probability __________ cannot occur, whereas an event with probability ___________ is certain to occur.

0,1,1

An event with probability greater than ____________ and less than ___________ involves uncertainty, and the closer its probability is to ___________, the more likely it is to occur.

mean, standard deviation

Any particular normal distribution is specified by its _____________ and ________________ ______________.

random variable

Associates a numeric value with each possible outcome in a situation involving uncertainty

d. 0.7745.

Given that Z is a standard normal random variable, P(-1.0 ≤ Z ≤ 1.5) is a. 0.0919. b. 0.8413. c. 0.9332. d. 0.7745.

b. -0.65.

Given that Z is a standard normal variable, the value z for which P(Z ≤ z) = 0.2580 is a. 0.70. b. -0.65. c. 0.242. d. 0.758.

a. 0.1498.

Given that the random variable X is normally distributed with a mean of 80 and a standard deviation of 10, P(85 ≤ X ≤ 90) is a. 0.1498. b. 0.1915. c. 0.5328. d. 0.3413.

a. 0.32

If A and B are any two events with P(A) = 0.8 and P(B|A) = 0.4, then the joint probability of A and B is: a. 0.32 b. 0.40 c. 1.20 d. 0.80

b. 0.70.

If A and B are mutually exclusive events with P(A) = 0.30 and P(B) = 0.40, then the probability that either A or B occur is a. 0.10. b. 0.70. c. 0.12. d. None of these choices are correct

b. cannot be larger than 0.30.

If A and B are mutually exclusive events with P(A) = 0.70, then P(B) a. can be any value between 0 and 0.70. b. cannot be larger than 0.30. c. can be any value between 0.30 and 0.70. d. can be any value between 0 and 1

b. 30.

If X is a normal random variable with a standard deviation of 10, then 3X has a standard deviation equal to a. 90. b. 30. c. 13. d. 10.

left

If Z is negative, the original value is the ______________ of the mean

right

If Z is positive, the original value is to the __________ of the mean

b. False

If events A and B have nonzero probabilities, then they can be both independent and mutually exclusive. a. True b. False

without replacement

If sampling is done _____________ _____________, each member of the population can be sampled only once. That is, once a person is sampled, his or her name is struck from the list and cannot be sampled again. applies to most real world sampling

with replacement

If sampling is done _____________ ______________, then it is possible, although maybe not likely, to select a given member of the population more than once. binomial model

a. True

If the random variable X is normally distributed with mean μ and standard deviation σ, then the random variable Z defined by Z = (X - μ)/σ is also normally distributed with mean 0 and standard deviation 1. a. True b. False

b. To the right of the mean

If the value of the standard normal random variable Z is positive, then the original score is where in relationship to the mean? a. To the left of the mean b. To the right of the mean c. Equal to the mean d. None of these choices are correct.

d. This cannot be determined from the information given.

If two events are independent, what is the probability that they both occur? a. 0.50 b. 0 c. 1.00 d. This cannot be determined from the information given.

a. 0.0

If two events are mutually exclusive and collectively exhaustive, what is the probability that both occur? a. 0.0 b. 0.5 c. 1.0 d. This can be any probability between 0 and 1.

d. This cannot be determined from the information given

If two events are mutually exclusive, what is the probability that one or the other occurs? a. 1.00 b. 0.25 c. 0.50 d. This cannot be determined from the information given

conditional probability

Let A and B be any events with probabilities P(A) and P(B). Typically, the probability P(A) is assessed without knowledge of whether B occurs. However, if you are told that B has occurred, then the probability of A might change. The new probability of A is called the _________ _________ of A given B, and it is denoted by ‍P(A|B)

d. mutually exclusive.

Let A and B be the events of the FDA approving and rejecting a new drug to treat hypertension, respectively. The events A and B are a. independent. b. conditional. c. unilateral. d. mutually exclusive.

multiplication rule

P (A and B) = P(A|B)P(B)

probabilistically independent events

P(A and B) = P(A)P(B)

probabilistically independent events

There are situations where the probabilities P(A), P(A|B), and P(A|B ̅) are equal. In this case, A and B are _____________ ____________ ___________. This does not mean that they are mutually exclusive. Rather, it means that knowledge of one event is of no value when assessing the probability of the other.

standardize

To _____________ a variable, subtract its mean and then divide the difference by the standard deviation. allows measuring variables with different means and/or standard deviations on a single scale.

variability

To measure the __________________ in a distribution, we calculate its variance or standard deviation.

mutually exclusive

When the events are _____________ _______________, the probability that at least one of the events will occur is the sum of their individual probabilities.

d. Z = (X - μ)/σ

Which equation shows the process of standardizing? a. E(X) = np b. f(x) = 1 - (σ/μ) c. E(Y) = μ d. Z = (X - μ)/σ

b. The sum of all probabilities for a random variable must be equal to 1.

Which of the following statements is true? a. Probabilities must be negative. b. The sum of all probabilities for a random variable must be equal to 1. c. Probabilities must be greater than 1. d. The sum of all probabilities for a random variable must be equal to 0.

Uncertainty, Risk

You typically have no control over __________________; it is something that simply exists. In contrast, ________________ depends on your position.

binomial distribution

a discrete distribution that can occur in two situations: When sampling from a population with only two types of members (males and females, for example) When performing a sequence of identical experiments, each of which has only two possible outcomes

Poisson distribution

a discrete distribution. It usually applies to the number of events occurring within a specified period of time or space.

probability

a number between 0 and 1 that measures the likelihood that some event will occur.

variance

denoted by σ2 or Var(X), is a weighted sum of the squared deviations of the possible values from the mean, where the weights are again the probabilities.

standard normal distribution

distribution that has mean 0 and standard deviation 1, so it is denoted by N(0,1). It is also referred to as the Z distribution.

Addition rule of probability

involves the probability that at least one of the events will occur.

probability distribution

lists all of the possible values of the random variable and their corresponding probabilities.

Z-value

number of standard deviations to the right or left of the mean.

discrete random variable

random variable that has only a finite number of possible values.

Normal distribution

single most important distribution in statistics. continuous distribution and the basis of the familiar symmetric bell-shaped curve. possible values ranging over the entire number line—from "minus infinity" to "plus infinity."

density function

usually denoted by f(x), specifies the probability distribution of a continuous random variable X. The higher f(x) is, the more likely x is. The total area between the graph of f(x) and the horizontal axis, which represents the total probability, is equal to 1. f(x) is nonnegative for all possible values of X. Probabilities are found from a density function as areas under the curve.

a. True

A binomial distribution with n number of trials, and probability of success p can be approximated well by a normal distribution with mean np and variance σ2 = np (1 - p) if np > 5 and n(1-p) > 5. a. True b. False

a. True

For a given probability of success p that is not too close to 0 or 1, the binomial distribution takes on more of a symmetric bell shape as the number of trials n increases. a. True b. False

Rule of Complements

P(Abar) = 1 - P(A)

c. discrete and continuous.

There are two types of random variables, they are a. real and unreal. b. exhaustive and mutually exclusive. c. discrete and continuous. d. complementary and cumulative.

b. False

Using the standard normal curve, the Z- score representing the 10th percentile is 1.28. a. True b. False

a. True

Using the standard normal curve, the Z- score representing the 75th percentile is 0.674. a. True b. False

a. True

Using the standard normal distribution, the Z- score representing the 99th percentile is 2.326. a. True b. False

b. False

Using the standard normal distribution, the Z-score representing the 5th percentile is 1.645. a. True b. False

exhaustive events

means they exhaust all possibilities—one of the events must occur.

mean

often denoted μ, is a weighted sum of the possible values, weighted by their probabilities.

Objective probabilities

probabilities that can be estimated from long-run proportions

cumulative probability

probability that the random variable is less than or equal to some particular value.

subjective probabilities

probability where one person's assessment of the likelihood that a certain event will occur.

continuous random variable

random variable has a continuum of possible values.

relative frequency

the proportion of times the event occurs out of the number of times the random experiment is run.

conditional probability

‍P(A|B) = P(A and B) / P(B)

discrete distribution

Distribution that results from a count

standard deviation for binomial distribution

Stdev(X) = Sqrt((np)(1-p))

a. True

Suppose that after graduation, you will either buy a new car (event A) or take a trip to Europe (event B). In this case, events A and B are mutually exclusive. a. True b. False

b. False

The Poisson probability distribution is one of the most commonly used continuous probability distributions. a. True b. False

a. a discrete random variable.

The binomial probability distribution is used with a. a discrete random variable. b. a continuous random variable. c. either a discrete or a continuous random variable, depending on the variance. d. either a discrete or a continuous random variable, depending on the sample size.

d. 16

The variance of a binomial distribution for which n = 100 and p = 0.20 is a. 20. b. 80. c. 100. d. 16.

right or left

By changing the mean, the normal curve shifts to the _________ or ___________

spread out

By changing the standard deviation, the curve becomes more or less ________ ____________.

continuous distribution

Distribution that results from a measurement

mean for binomial distribution

E(X) = np

mutually exclusive

Events are ____________ __________ if at most one of them can occur—that is, if one of them occurs, then none of the others can occur.

conditional mean and variance

There are many situations where the mean and variance of a random variable depend on some external event.

a. rule of complements.

P(Ā) = 1 - P(A) is the a. rule of complements. b. rule of opposites. c. commutative rule. d. addition rule.

The probabilities must be nonnegative. They must sum to 1.

Probability distributions must satisfy two criteria:

b. each member of the population can be sampled only once.

Sampling done without replacement means that a. only certain members of the population can be sampled. b. each member of the population can be sampled only once. c. each member of the population can be sampled twice. d. each member of the population can be sampled repeatedly.

d. conditional

The formal way to revise probabilities based on new information is to use _____ probabilities. a. unilateral b. common sense c. complementary d. conditional

d. Poisson distribution

The local police department is interested in estimating the number of cars that fail to stop at a stop sign during a specified lunch hour. Which probability distribution should they use? a. Normal distribution b. Uniform distribution c. Binomial distribution d. Poisson distribution

b. False

The mean and standard deviation of a normally distributed random variable that has been "standardized" are zero and one, respectively. a. True b. False

expected value

The mean is also called the ___________ ____________ of x and denoted E(X).

c. central location.

The mean μ of a probability distribution is a measure of a. variability of the distribution. b. the shape of the distribution. c. central location. d. skewness of the distribution.

d. continuous distribution with two parameters.

The normal distribution is a a. discrete distribution with two parameters. b. binomial distribution with only one parameter. c. density function of a discrete random variable. d. continuous distribution with two parameters.

two-parameter family

The normal distribution is a ______ - __________ _________, where the two parameters are the mean and standard deviation.

a. True

The number of car insurance policy holders is an example of a discrete random variable. a. True b. False

b. different means and standard deviations on a single scale.

The primary reason for standardizing random variables is to measure variables with a. dissimilar means and standard deviations in like terms. b. different means and standard deviations on a single scale. c. different means and standard deviations on a non-standard scale. d. similar means and standard deviations on two scales.

c. 0.35.

The probabilities shown in a table with two rows, A1 and A2 and two columns, B1 and B2, are as follows: P(A1 and B1) = 0.10, P(A1 and B2) = 0.30, P(A2 and B1) = 0.05, and P(A2 and B2) = 0.55. Then P(A1|B2) is a. 0.65. b. 0.67. c. 0.35. d. 0.33

d. variability of the distribution.

The standard deviation σ of a probability distribution is a measure of a. skewness of the distribution. b. central location. c. the shape of the distribution. d. variability of the distribution.

a. 0 and 1.

The standard normal distribution has a mean and a standard deviation respectively equal to a. 0 and 1. b. 0 and 0. c. 1 and 1. d. 1 and 0

a. True

The temperature of the room in which you are writing this test is a continuous random variable. a. True b. False

a. True

The total area under the normal distribution curve is equal to one. a. True b. False