Question 1

Larger values of the standard deviation result in a normal curve that is
A)shifted to the right
B)shifted to the left
C)narrower and more peaked
D)wider and flatter

Accepted Answer

A larger standard deviation means that the data points are more spread out from the mean. This results in a wider and flatter normal curve, as opposed to a narrower and more peaked curve with a smaller standard deviation.

Question 2

A probability density function
A)gives the probability that the random variable equals a specific value of x.
B)is used for discrete random variables.
C)describes the graph whose underlying area represents probability over an interval.
D)each of these choices are true.

Accepted Answer

A probability density function describes the graph whose underlying area represents the probability over an interval, but it does not give the probability that the random variable equals a specific value of x (that would be the probability mass function). Probability density functions are used for continuous random variables, not discrete ones.

Question 3

Experiments with repeated independent trials will be described by the binomial distribution if
A)the trials are continuous.
B)each trial result influences the next.
C)the time between trials is constant.
D)each trial has exactly two outcomes whose probabilities do not change.

Accepted Answer

The binomial distribution applies to experiments with repeated independent trials where each trial has exactly two outcomes (success or failure) with fixed probabilities that do not change. The other options do not meet these criteria.

Question 4

The uniform distribution defined over the interval from 25 to 40 has the probability density function
A)f(x) = 1/40 for all x
B)f(x) = 5/8 for 25 F1F1F1S1F1F1F10 x F1F1F1S1F1F1F10 40 and f(x) = 0 elsewhere
C)f(x) = 1/25 for 0 F1F1F1S1F1F1F10 x F1F1F1S1F1F1F10 25 and f(x) = 1/40 for 26 F1F1F1S1F1F1F10 x F1F1F1S1F1F1F10 40
D)f(x) = 1/15 for 25 F1F1F1S1F1F1F10 x F1F1F1S1F1F1F10 40 and f(x) = 0 elsewhere

Accepted Answer

The probability density function for a uniform distribution over an interval [a, b] is f(x) = 1/(b-a) for a ≤ x ≤ b. Here, a = 25 and b = 40, so f(x) = 1/(40-25) = 1/15 for 25 ≤ x ≤ 40.

Question 5

In order to measure the dispersion of a random variable,look at its
A)standard deviation.
B)mean.
C)expected value.
D)average.

Accepted Answer

The standard deviation is the most common measure of dispersion for a random variable. It measures how spread out the values of the variable are from the mean, indicating the amount of variability in the data. The other options (mean, expected value, average) are measures of central tendency, not dispersion.

Question 6

The weight of an object,measured to the nearest gram,is an example of
A)a continuous random variable
B)a discrete random variable
C)either a continuous or a discrete random variable, depending on the weight of the object
D)either a continuous or a discrete random variable depending on the units of measurement

Accepted Answer

The weight of an object measured to the nearest gram is an example of a discrete random variable because it can only take on specific, separate values (e.g., 1 gram, 2 grams, etc.) and not any value in between.

Question 7

When the exponential distribution probability is given as P(x F1F1F1S1F1F1F10 9 )= 1 F1F1F1S1F1F1F10 eF1F1F1S1^F1F1F10^9/18 the average value for x is A)18 B)9 C)9/18 D)1/2

Accepted Answer

The exponential distribution has the probability density function P(x ≤ a) = 1 - e^(-a/λ), where λ is the average value or mean. Given P(x ≤ 9) = 1 - e^(-9/18), λ is 18.

Question 8

z is a standard normal random variable.The P(z F1F1F1S1F1F1F10 2.11)equals
A)0.4821
B)0.9821
C)0.5
D)0.0174

Accepted Answer

Using a standard normal distribution table, we find that P(z ≥ 2.11) is approximately 0.0174. This corresponds to the probability of getting a value that is 2.11 standard deviations or more above the mean of the standard normal distribution.

Question 9

The Poisson probability distribution is used with
A)a discrete random variable
B)a continuous random variable
C)either a discrete or continuous random variable
D)any random variable

Accepted Answer

The Poisson probability distribution is only used with a discrete random variable. It describes the probability distribution of the number of events that occur in a specific time period or space, where the events occur independently and with a constant mean rate. The Poisson distribution function is only defined for discrete, non-negative values of a random variable. Therefore, choice A is the correct answer.

Question 10

The binomial probability distribution is most symmetric when
A)n is 30 or greater
B)n equals p
C)p approaches 1
D)p equals 0.5

Accepted Answer

The binomial distribution is most symmetric when p equals 0.5, meaning there is an equal probability of success and failure. This is because the probability of getting k successes out of n trials is equal to the probability of getting n-k failures out of n trials. As p deviates from 0.5, the distribution becomes more skewed. Therefore, D is the best choice.

Question 11

An experiment consists of measuring the speed of automobiles on a highway by the use of radar equipment.The random variable in this experiment is speed,measured in miles per hour.This random variable is a
A)discrete random variable
B)continuous random variable
C)complex random variable
D)None of these alternatives is correct.

Accepted Answer

The speed of automobiles on a highway can take on any real value between 0 mph and the maximum speed limit, making it a continuous random variable.

Question 12

A numerical description of the outcome of an experiment is
A)a normal variable.
B)a discrete variable.
C)a random variable.
D)an experimental variable.

Accepted Answer

A numerical description of the outcome of an experiment is a random variable, as it can take on different values with varying probabilities. Option A, a normal variable, refers to a specific distribution shape and does not apply to all numerical outcomes. Option B, a discrete variable, refers to a variable that can only take on certain values, which may not be the case for all numerical outcomes. Option D, an experimental variable, refers to a variable that is intentionally manipulated in an experiment, which is not necessary for all numerical outcomes.

Question 13

A statement that matches the values of a random variable with the probabilities of those values is
A)the expected value.
B)the variation of the random variable.
C)an experiment.
D)a probability distribution.

Accepted Answer

A probability distribution is a function that maps the values of a random variable to their corresponding probabilities. This statement describes the process of matching values with probabilities, which is the definition of a probability distribution. The other answer choices are not relevant to this concept.

Question 14

Which of the following are continuous random variables?
I.the weight of an elephant
II.the time to answer a questionnaire
III.the number of floors in a skyscraper
IV.the square feet of countertop in a kitchen
A)I and II only
B)III and IV only
C)I, II and IV
D)I, II, II, and IV

Accepted Answer

Continuous random variables can take any value within a range. The weight of an elephant, the time to answer a questionnaire, and the square feet of countertop in a kitchen are all measurements that can vary continuously. The number of floors in a skyscraper is a discrete variable, as it can only take whole number values.

Question 15

A standard normal distribution is a normal distribution with
A)a mean of 1 and a standard deviation of 0
B)a mean of 0 and a standard deviation of 1
C)any mean and a standard deviation of 1
D)any mean and any standard deviation

Accepted Answer

A standard normal distribution has a mean of 0 and a standard deviation of 1, which is represented by the symbol "Z". This means that the values of the distribution are measured in terms of the number of standard deviations away from the mean.

Question 16

If x is normally distributed with mean 12 and standard deviation 2,then P(x F1F1F1S1F1F1F10 9)is
A)P(z F1F1F1S1F1F1F10 9/10).
B)P(z F1F1F1S1F1F1F10 F1F1F1S1F1F1F103/2)
C)P(z F1F1F1S1F1F1F10 2/3)
D)P(z F1F1F1S1F1F1F10 F1F1F1S1F1F1F103/4)

Accepted Answer

To find P(x ≤ 9), we convert x to a standard normal variable z using z = (x - mean) / standard deviation. Here, z = (9 - 12) / 2 = -3/2. Thus, the correct choice is B.

Question 17

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)Poisson probability distribution
D)normal probability distribution

Accepted Answer

The characteristics described in the question (fixed probability of success, count of successes in a fixed number of trials) are the defining features of a binomial probability distribution.

Question 18

The expected value for a binomial probability distribution is
A)E(x) F1F1F1S1F1F1F10 np
B)E(x) F1F1F1S1F1F1F10 p(1 F1F1F1S1F1F1F10 p)
C)E(x) F1F1F1S1F1F1F10 np(1 F1F1F1S1F1F1F10 p)
D)E(x) F1F1F1S1F1F1F10 np(1 F1F1F1S1F1F1F10 n)

Accepted Answer

The expected value for a binomial probability distribution is E(x) = np.

Question 19

When dealing with the number of occurrences of an event over a specified interval of time or space and when the occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval,the appropriate probability distribution is a
A)binomial distribution
B)Poisson distribution
C)normal distribution
D)hypergeometric distribution

Accepted Answer

When dealing with the number of occurrences of an event over a specified interval of time or space and when the occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval, the appropriate probability distribution is a Poisson distribution. The Poisson distribution is used to model events that occur independently and at a constant rate within a certain time or space interval. Therefore, it is the best choice for this scenario.

Question 20

In a Poisson probability problem,the rate of errors is one every two hours.To find the probability of three defects in four hours,
A)F1F1F1S1F1F1F10 = 1, x = 4
B)F1F1F1S1F1F1F10 = 2, x = 3
C)F1F1F1S1F1F1F10 = 3, x = 2
D)F1F1F1S1F1F1F10 = 3, x = 6

Accepted Answer

The rate of errors is 1 every 2 hours, so in 4 hours, the rate (λ) is 2. We want the probability of 3 defects (x = 3).

Quiz 3: Probability Distributions

Larger values of the standard deviation result in a normal curve that is

A probability density function

Experiments with repeated independent trials will be described by the binomial distribution if

The uniform distribution defined over the interval from 25 to 40 has the probability density function

In order to measure the dispersion of a random variable,look at its

The weight of an object,measured to the nearest gram,is an example of

When the exponential distribution probability is given as P(x  9 ) = 1  e^^9/18 the average value for x is

z is a standard normal random variable.The P(z  2.11) equals

The Poisson probability distribution is used with

The binomial probability distribution is most symmetric when

An experiment consists of measuring the speed of automobiles on a highway by the use of radar equipment.The random variable in this experiment is speed,measured in miles per hour.This random variable is a

A numerical description of the outcome of an experiment is

A statement that matches the values of a random variable with the probabilities of those values is

Which of the following are continuous random variables? I.the weight of an elephant II.the time to answer a questionnaire III.the number of floors in a skyscraper IV.the square feet of countertop in a kitchen

A standard normal distribution is a normal distribution with

If x is normally distributed with mean 12 and standard deviation 2,then P(x  9) 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

The expected value for a binomial probability distribution is

When dealing with the number of occurrences of an event over a specified interval of time or space and when the occurrence or nonoccurrence in any interval is independent of the occurrence or nonoccurrence in any other interval,the appropriate probability distribution is a

In a Poisson probability problem,the rate of errors is one every two hours.To find the probability of three defects in four hours,