Question 1

With two random variables x and y,a positive covariance says that as x increases,y tends to increase in a linear fashion.

Accepted Answer

A positive covariance between x and y indicates that when x increases, y tends to increase as well, suggesting a linear relationship between the two variables.

Question 2

If the population size is at least 20 times larger than the sample size,a hypergeometric distribution can be approximated by the binomial distribution.

Accepted Answer

This is known as the large population approximation or the finite population correction. When the sample size is less than 5% of the population size, the hypergeometric distribution can be approximated by the binomial distribution.

Question 3

Depending on the mean of the Poisson distribution,the distribution can either be very skewed to the right or quite symmetrical.

Accepted Answer

The Poisson distribution is skewed to the right when the mean is small, but as the mean increases, the distribution becomes more symmetrical and approaches a normal distribution.

Question 4

A correlation coefficient is a unitless measure of the linear relationship between two random variables.

Accepted Answer

A correlation coefficient is a unitless measure used to determine the degree to which two variables are linearly related. It ranges from -1 to 1, with -1 indicating a strong negative correlation, 1 indicating a strong positive correlation, and 0 indicating no correlation.

Question 5

In a binomial distribution,the random variable X is continuous.

Accepted Answer

In a binomial distribution, the random variable X is discrete, with a fixed number of trials and two possible outcomes for each trial.

Question 6

If the number of surface nonconformities on a specific size of a metal piece is the discrete random variable in question,then the appropriate probability distribution that can describe the probability of a specific size metal sheet containing 3 nonconformities is most likely given by the binomial distribution.

Accepted Answer

This example is a description of a hypergeometric distribution.

Question 7

In a hypergeometric probability distribution of a population of N items,r refers to the number of successes and N − r to the number of failures.

Accepted Answer

This is true according to the definition of a hypergeometric probability distribution.

Question 8

The standard deviation of a discrete random variable measures the spread of the population of all possible values of x.

Accepted Answer

This is true. The standard deviation is a measure of how spread out the data is from the mean, so it can be used to determine how much variability there is among the possible values of a random variable.

Question 9

The internal auditor for your company believes that 10 percent of their invoices contain errors.To check this theory,20 invoices are randomly selected and 5 are found to have errors.The claim of the auditor will be rejected.

Accepted Answer

The auditor's claim is rejected because the sample of 20 invoices found a higher percentage of errors (5/20 = 25%) than the auditor's claim of 10%.

Question 10

The time (in seconds)it takes for an athlete to run 50 meters is an example of a continuous random variable.

Accepted Answer

A continuous random variable can take on any value within a range (in this case, the time it takes to run 50 meters), including decimal values.

Question 11

The variable Home Ownership can take on one of two values,1 if the person living in a home owns the home and 0 if the person living in a home does not own the home.This is an example of a discrete random variable.

Accepted Answer

This statement is true. Home Ownership can only take on two distinct values, 1 or 0, making it a discrete random variable.

Question 12

The mean and the variance of a Poisson random variable are equal.

Accepted Answer

This statement is true. The mean of a Poisson random variable is equal to its variance, which is lambda, the parameter of the Poisson distribution.

Question 13

The binomial experiment consists of n independent,identical trials,each of which results in either success or failure and is such that the probability of success on any trial is the same.

Accepted Answer

This is the definition of a binomial experiment.

Question 14

A Poisson random variable is a continuous variable that can be used to describe the number of occurrences of an event over a specified interval of time or space.

Accepted Answer

A Poisson random variable is actually a discrete variable, not a continuous one. It can be used to describe the number of occurrences of an event in a fixed interval of time or space, but it only takes on integer values, not continuous values.

Question 15

The mean of the binomial distribution is np(1 − p).

Accepted Answer

The mean of a binomial distribution is given by np, where n is the number of trials and p is the probability of success.

Question 16

The expected value of the discrete random variable x is the population mean.

Accepted Answer

The expected value of a discrete random variable x is defined as the sum of the product of each possible value of x and its corresponding probability. This is equivalent to the population mean, so the statement is true.

Question 17

The hypergeometric probability distribution can be approximated by the Poisson distribution.

Accepted Answer

The hypergeometric distribution is approximated by a binomial distribution.

Question 18

In a binomial experiment,the results of one trial are dependent on the results of other trials.

Accepted Answer

In a binomial experiment, each trial is independent of the previous trials.

Question 19

For a discrete probability distribution,the value of p(x)for each value of x falls between −1 and 1.

Accepted Answer

Probability values, p(x), can only fall between 0 and 1.

Question 20

A discrete random variable may assume a countable number of outcome values.

Accepted Answer

This is the definition of a discrete random variable.

Quiz 5: Discrete Random Variables