Question 1

The variance is a weighted average of the
A)square root of the deviations from the mean
B)square root of the deviations from the median
C)squared deviations from the median
D)squared deviations from the mean

Accepted Answer

The variance is computed as the average of the squared deviations from the mean of a set of data. It is an important measure of dispersion that reflects how much the data values are spread out from the mean. The other options are incorrect because the square root of the deviations from the mean or median would yield the standard deviation, not the variance, and the squared deviations from the median would not be an appropriate measure of variability for a dataset with a mean.

Question 2

The expected value of a discrete random variable
A)is the most likely or highest probability value for the random variable
B)will always be one of the values x can take on, although it may not be the highest probability value for the random variable
C)is the average value for the random variable over many repeats of the experiment
D)All of the answers are correct.

Accepted Answer

The expected value of a discrete random variable is the average value for the random variable over many repeats of the experiment. It takes into account all possible outcomes weighted by their respective probabilities, and is typically denoted by E(X) for a random variable X. Although the expected value may not necessarily be the most likely or highest probability value for the random variable, it provides a useful measure of central tendency and is often used in statistical analysis and decision-making.

Question 3

Which of the following statements about a discrete random variable and its probability distribution are true?
A)Values of the random variable can never be negative.
B)Negative values of f(x) are allowed as long as  $\Sigma$f(x) = 1.
C)Values of f(x) must be greater than or equal to zero.
D)The values of f(x) increase to a maximum point and then decrease.

Accepted Answer

This statement is true because the probability of any outcome cannot be less than zero. A probability distribution function f(x) gives the probability of each possible value or outcome x of a discrete random variable. The sum of probabilities of all possible outcomes is equal to 1. Negative values of f(x) are not allowed because it would indicate a negative probability, which is not possible. The values of f(x) can take any non-negative value, including zero. The shape of the probability distribution function can vary depending on the specific distribution, but it does not necessarily increase to a maximum point and then decrease.

Question 4

An experiment consists of making 80 telephone calls in order to sell a particular insurance policy. The random variable in this experiment is the number of sales made. This random variable is a
A)discrete random variable
B)continuous random variable
C)complex random variable
D)None of the answers is correct.

Accepted Answer

The number of sales made can only take on specific values (0, 1, 2, 3, ...), making it a discrete random variable.

Question 5

A numerical description of the outcome of an experiment is called a
A)descriptive statistic
B)probability function
C)variance
D)random variable

Accepted Answer

A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. It is a way to assign numbers to outcomes of an experiment, thus providing a numerical description of those outcomes.

Question 6

The number of customers that enter a store during one day 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 number of the customers
D)either a continuous or a discrete random variable, depending on the gender of the customers

Accepted Answer

The number of customers that enter a store during one day is a discrete random variable because it can only take on whole number values (e.g. 0, 1, 2, 3, etc.). Additionally, it is unlikely that the number of customers would take on a fractional value. Therefore, it is not a continuous random variable. The number of customers entering the store does not depend on gender, so option D is not applicable.

Question 7

Variance is
A)a measure of the average, or central value of a random variable
B)a measure of the dispersion of a random variable
C)the square root of the standard deviation
D)the sum of the deviation of data elements from the mean

Accepted Answer

Variance is a measure of how spread out the data is. It measures the deviation of each number in the dataset from the mean and then squaring those deviations to find the average. Therefore, the correct choice is B - variance is a measure of the dispersion of a random variable.

Question 8

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

A discrete random variable takes on a countable number of possible values, and in this case, the weight of an object can only be measured to a finite number of grams, making it a discrete random variable.

Question 9

The expected value of a random variable is the
A)value of the random variable that should be observed on the next repeat of the experiment
B)value of the random variable that occurs most frequently
C)square root of the variance
D)None of the answers is correct.

Accepted Answer

The expected value of a random variable is a theoretical mean of all possible outcomes, weighted by their probabilities, not necessarily any of the options given. It is not specifically the value observed next, the most frequent value, or the square root of the variance.

Question 10

A continuous random variable may assume
A)any value in an interval or collection of intervals
B)only integer values in an interval or collection of intervals
C)only fractional values in an interval or collection of intervals
D)only the positive integer values in an interval

Accepted Answer

A continuous random variable can assume any value in an interval or collection of intervals. This is because the variable can take on any value within its range, which is a continuous set of values. Examples of continuous random variables include height, weight, and time.

Question 11

The weight of an object, measured in grams, 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 in grams, can take on any value within a range (e.g. 1 gram, 1.5 grams, 2 grams, etc.), making it a continuous random variable.

Question 12

Which of the following is not a required condition for a discrete probability function?
A)f(x) $\ge$ 0 for all values of x
B) $\Sigma$f(x) = 1
C) $\Sigma$f(x) = 0
D)All of the answers are correct.

Accepted Answer

The sum of the probabilities for all possible values of x must equal 1 (option B), and the probability for each possible value of x must be between 0 and 1 (option A), but there is no requirement that the sum of the probabilities for all possible values of x must equal 0.

Question 13

Excel's __________ function can be used to compute the expected value of a discrete random variable.
A)SUMPRODUCT
B)AVERAGE
C)MEDIAN
D)VAR

Accepted Answer

The SUMPRODUCT function in Excel can be used to compute the expected value of a discrete random variable by multiplying each possible value of the variable by its corresponding probability and then summing all these products.

Question 14

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 the answers is correct.

Accepted Answer

Speed is a continuous variable, as it can take on any value within a range (e.g. 55.3 mph, 65.8 mph, etc.). Therefore, the random variable in this experiment is a continuous random variable.

Question 15

A measure of the average value of a random variable is called a(n)
A)variance
B)standard deviation
C)expected value
D)None of the answers is correct.

Accepted Answer

The expected value of a random variable is the average value that we expect to see. Variance and standard deviation are measures of the spread or variability of the random variable, not the average value.

Question 16

A description of how the probabilities are distributed over the values the random variable can assume is called a
A)probability distribution
B)probability function
C)random variable
D)expected value

Accepted Answer

A probability distribution describes how probabilities are distributed over the possible values of a random variable. Option B is incorrect because a probability function is a specific type of probability distribution that maps each possible outcome to a probability. Option C is incorrect because a random variable is a variable whose possible values are determined by chance. Option D is incorrect because expected value is a concept used in probability theory to describe the long-term average value of a random variable.

Question 17

Which of the following is(are) required condition(s) for a discrete probability function?
A) $\Sigma$f(x) = 0
B)f(x) $\ge$ 1 for all values of x
C)f(x) < 0
D)None of the answers is correct.

Accepted Answer

A discrete probability function must satisfy two conditions: 1) The sum of all probabilities must equal 1, not 0. 2) Each probability must be between 0 and 1, inclusive, meaning it cannot be negative or greater than 1. None of the given options correctly describe these conditions.

Question 18

A random variable that can assume only a finite number of values is referred to as a(n)
A)infinite sequence
B)finite sequence
C)discrete random variable
D)discrete probability function

Accepted Answer

A random variable that can assume only a finite number of values is called a discrete random variable. Option A and B are incorrect because they do not specifically refer to the limited number of values. Option D is also incorrect because a discrete probability function is a way to describe the probability distribution of a discrete random variable, not the variable itself.

Question 19

Excel's __________ function can be used to compute the variance of a discrete random variable.
A)SUMPRODUCT
B)AVERAGE
C)MEDIAN
D)VAR

Accepted Answer

SUMPRODUCT function can be used to compute the variance of a discrete random variable by multiplying each value by its corresponding probability and then summing up these products.

Question 20

A weighted average of the value of a random variable, where the probability function provides weights is known as
A)a probability function
B)a random variable
C)the expected value
D)None of the answers is correct

Accepted Answer

The weighted average of a random variable is known as the expected value, which uses the probability function as weights. Therefore, the correct answer is C, "the expected value."

Quiz 5: Discrete Probability Distributions

The variance is a weighted average of the

The expected value of a discrete random variable

Which of the following statements about a discrete random variable and its probability distribution are true?

An experiment consists of making 80 telephone calls in order to sell a particular insurance policy. The random variable in this experiment is the number of sales made. This random variable is a

A numerical description of the outcome of an experiment is called a

The number of customers that enter a store during one day is an example of

Variance is

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

The expected value of a random variable is the

A continuous random variable may assume

The weight of an object, measured in grams, is an example of

Which of the following is not a required condition for a discrete probability function?

Excel's __________ function can be used to compute the expected value of a discrete random variable.

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 measure of the average value of a random variable is called a(n)

A description of how the probabilities are distributed over the values the random variable can assume is called a

Which of the following is(are) required condition(s) for a discrete probability function?

A random variable that can assume only a finite number of values is referred to as a(n)

Excel's __________ function can be used to compute the variance of a discrete random variable.

A weighted average of the value of a random variable, where the probability function provides weights is known as