Question 1

It is against the law to discriminate against job applicants because of race, religion, sex, or age. Of the individuals who apply for an accountant's position in a large corporation, 39% are over 45 years old. If the company decides to choose 96 of a very large number of applicants for closer credential screening, claiming that the selection will be random and not age-biased, what is the z-value associated with fewer than 46 of those chosen being over 45 years old? (Assume that the applicant pool is large enough so that x, the number in the sample over 45 years old, has a binomial probability distribution.)

Accepted Answer

The answer of It is against the law to discriminate...

Question 2

Transportation officials tell us that 80% of drivers wear seat belts while driving. What is the probability of observing 633 or fewer drivers wearing seat belts in a sample of 850 drivers?

Accepted Answer

The probability of observing 633 or fewer drivers wearing seat belts in a sample of 850 drivers is approximately 0. This is because the probability of observing 633 or fewer drivers wearing seat belts is the same as the probability of observing 634 or more drivers not wearing seat belts. The probability of observing 634 or more drivers not wearing seat belts is very small, since 80% of drivers wear seat belts.

Question 3

A certain baseball player hits a home run in 4% of his at-bats. Consider his at-bats as independent events. Find the probability that this baseball player hits at most 16 home runs in 650 at-bats?

Accepted Answer

The probability of hitting at most 16 home runs in 650 at-bats can be calculated using the binomial distribution with n = 650 and p = 0.04. The cumulative probability for hitting 16 or fewer home runs is approximately 0.0287.

Question 4

A certain baseball player hits a home run in 6% of his at-bats. Consider his at-bats as independent events. Find the probability that this baseball player hits more than 32 home runs in 750 at-bats?

Accepted Answer

The probability of hitting a home run is 0.06. The probability of not hitting a home run is 0.94. The probability of hitting more than 32 home runs in 750 at-bats is the complement of the probability of hitting 32 or fewer home runs in 750 at-bats. The probability of hitting 32 or fewer home runs in 750 at-bats is the sum of the probabilities of hitting 0, 1, 2, ..., 32 home runs in 750 at-bats. This sum can be calculated using the binomial distribution. The probability of hitting 0 home runs in 750 at-bats is (0.94)^750 = 0.0000. The probability of hitting 1 home run in 750 at-bats is 750 * (0.06) * (0.94)^749 = 0.0002. The probability of hitting 2 home runs in 750 at-bats is 750 * (0.06)^2 * (0.94)^748 = 0.0010. ... The probability of hitting 32 home runs in 750 at-bats is 750 * (0.06)^32 * (0.94)^718 = 0.0000. The sum of these probabilities is 0.0012. The complement of this probability is 1 - 0.0012 = 0.9988. Therefore, the probability of hitting more than 32 home runs in 750 at-bats is 0.9988.

Question 5

The exponential distribution is sometimes called the waiting-time distribution, because it is used to describe the length of time between occurrences of random events.

Accepted Answer

The exponential distribution is commonly used to model waiting times between occurrences of events in various fields, such as physics, engineering, and economics. Therefore, it is often referred to as the waiting-time distribution.

Question 6

A loan officer has 66 loan applications to screen during the next week. If past record indicates that she turns down 17% of the applicants, what is the z-value associated with 61 or more of the 66 applications being rejected?

Accepted Answer

The answer of A loan officer has 66 loan applications...

Question 7

Transportation officials tell us that 80% of drivers wear seat belts while driving. What is the probability that between 538 and 546 drivers in a sample of 700 drivers wear seat belts?

Accepted Answer

The probability of a driver wearing a seat belt is 0.8. The probability of a driver not wearing a seat belt is 0.2. The probability of between 538 and 546 drivers in a sample of 700 drivers wearing seat belts is the sum of the probabilities of 538, 539, 540, 541, 542, 543, 544, 545, and 546 drivers wearing seat belts. This can be calculated using the binomial distribution. The probability of 538 drivers wearing seat belts is 0.0837. The probability of 539 drivers wearing seat belts is 0.0166. The probability of 540 drivers wearing seat belts is 0.01003. The probability of 541 drivers wearing seat belts is 0.00501. The probability of 542 drivers wearing seat belts is 0.00200. The probability of 543 drivers wearing seat belts is 0.00071. The probability of 544 drivers wearing seat belts is 0.00024. The probability of 545 drivers wearing seat belts is 0.00008. The probability of 546 drivers wearing seat belts is 0.00002. The sum of these probabilities is 0.0837.

Question 8

Suppose that x has an exponential distribution with $\theta = 1.5 \text {. Find } P ( x > 1 )$

Accepted Answer

The probability density function of the exponential distribution is given by:$$f(x) = \theta e^{-\theta x}$$where $\theta$ is the rate parameter.In this case, $\theta = 1.5$, so the probability density function is:$$f(x) = 1.5 e^{-1.5 x}$$The probability that $x > 1$ is given by the integral of the probability density function from $1$ to infinity:$$P(x > 1) = \int_1^\infty 1.5 e^{-1.5 x} dx$$This integral evaluates to:$$P(x > 1) = 0.513417$$Therefore, the correct answer is A.

Question 9

A study of college students stated that 25% of all college students have at least one tattoo. In a random sample of 80 college students, let x be the number of the students that have at least one tattoo. Find the approximate probability that more than 30 of the sampled students had at least one tattoo.

Accepted Answer

The answer of A study of college students stated that...

Question 10

A study of college students stated that 25% of all college students have at least one tattoo. In a random sample of 80 college students, let x be the number of the students that have at least one tattoo. Find the approximate probability that more than 17 and less than 26 of the sampled students had at least one tattoo.

Accepted Answer

The mean is 20 (80 * 0.25) and the standard deviation is 3.87 (sqrt(80 * 0.25 * 0.75)). Using the normal approximation, calculate the z-scores for 17.5 and 25.5, then find the probability between these z-scores.

Question 11

A study of college students stated that 25% of all college students have at least one tattoo. In a random sample of 80 college students, let x be the number of the students that have at least one tattoo. Find the mean and standard deviation for this binomial distribution.

Accepted Answer

The mean of a binomial distribution is calculated as n * p, and the standard deviation is sqrt(n * p * (1-p)). Here, n = 80 and p = 0.25, so the mean is 80 * 0.25 = 20, and the standard deviation is sqrt(80 * 0.25 * 0.75) ≈ 3.87.

Question 12

The exponential distribution has the property that its mean equals its standard deviation.

Accepted Answer

This is true. The mean and standard deviation of an exponential distribution are both equal to 1/λ, where λ is the rate parameter.

Question 13

The exponential distribution is governed by two quantities, μ and σ, that determine its shape and location

Accepted Answer

The exponential distribution is actually governed by a single parameter, λ (or alternatively, the mean value μ = 1/λ). The parameter λ determines the rate at which events occur in the distribution. The standard deviation of the distribution is also equal to the mean, so there is no separate parameter for the dispersion or shape of the distribution.

Question 14

Assume that x is a binomial random variable with n = 1000 and p = 0.80. Use a normal approximation to find P(800 < x ≤ 820).

Accepted Answer

The correct answer is A because the normal approximation to the binomial distribution is accurate when n is large and p is not too close to 0 or 1. In this case, n = 1000 and p = 0.80, so the normal approximation is appropriate. The mean of the binomial distribution is np = 1000 * 0.80 = 800, and the standard deviation is sqrt(np(1-p)) = sqrt(1000 * 0.80 * 0.20) = 11.28. Therefore, the z-score for x = 800 is (800 - 800) / 11.28 = 0, and the z-score for x = 820 is (820 - 800) / 11.28 = 1.78. The probability of 0 < z < 1.78 is 0.4314, which is the answer.

Question 15

Transportation officials tell us that 80% of drivers wear seat belts while driving. Find the probability that more than 576 drivers in a sample of 700 drivers wear seat belts.

Accepted Answer

The probability of a driver wearing a seat belt is 0.8. The probability of a driver not wearing a seat belt is 0.2. The probability of more than 576 drivers in a sample of 700 drivers wearing seat belts is the same as the probability of less than 124 drivers in a sample of 700 drivers not wearing seat belts. This is because the sum of the probabilities of all possible outcomes must be 1. The probability of less than 124 drivers in a sample of 700 drivers not wearing seat belts is 0.0594.

Question 16

Suppose that 88% of the stocks listed on a particular exchange increased in value yesterday. Let x be the number of stocks that increased in value yesterday in a random of 72 stocks listed on the exchange. Find the mean and standard deviation of x.

Accepted Answer

The answer of Suppose that 88% of the stocks listed...

Question 17

Determine the value of $e ^ { - a / \theta } \text { for } \theta = 2 \text { and } a = 3$

Accepted Answer

The expression $e^{-a/\theta}$ becomes $e^{-3/2}$ when $\theta = 2$ and $a = 3$. Calculating $e^{-3/2}$ gives approximately 0.223130.

Question 18

The probability density function for an exponential random variable x has a graph called a bell curve.

Accepted Answer

The exponential distribution does not have a bell-shaped curve. It has a decreasing exponential distribution, which means it is skewed to the right. The bell-shaped curve is characteristic of the normal distribution.

Question 19

A certain baseball player hits a home run in 3% of his at-bats. Consider his at-bats as independent events. How many home runs do we expect the baseball player to hit in 600 at-bats?

Accepted Answer

The expected number of home runs is calculated by multiplying the probability of hitting a home run (3% or 0.03) by the number of at-bats (600), which equals 18.

Question 20

Suppose that 67% of the employees of a company participate in the company's medical savings program. Let x be the number of employees who participate in the program in a random sample of 50 employees. Find the mean and standard deviation of x.

Accepted Answer

The answer of Suppose that 67% of the employees of...

Quiz 5: Continuous Random Variables

Transportation officials tell us that 80% of drivers wear seat belts while driving. What is the probability of observing 633 or fewer drivers wearing seat belts in a sample of 850 drivers?

A certain baseball player hits a home run in 4% of his at-bats. Consider his at-bats as independent events. Find the probability that this baseball player hits at most 16 home runs in 650 at-bats?

A certain baseball player hits a home run in 6% of his at-bats. Consider his at-bats as independent events. Find the probability that this baseball player hits more than 32 home runs in 750 at-bats?

The exponential distribution is sometimes called the waiting-time distribution, because it is used to describe the length of time between occurrences of random events.

A loan officer has 66 loan applications to screen during the next week. If past record indicates that she turns down 17% of the applicants, what is the z-value associated with 61 or more of the 66 applications being rejected?

Transportation officials tell us that 80% of drivers wear seat belts while driving. What is the probability that between 538 and 546 drivers in a sample of 700 drivers wear seat belts?

Suppose that x has an exponential distribution with $\theta = 1.5 \text {. Find } P ( x > 1 )$

A study of college students stated that 25% of all college students have at least one tattoo. In a random sample of 80 college students, let x be the number of the students that have at least one tattoo. Find the mean and standard deviation for this binomial distribution.

The exponential distribution has the property that its mean equals its standard deviation.

The exponential distribution is governed by two quantities, μ and σ, that determine its shape and location

Assume that x is a binomial random variable with n = 1000 and p = 0.80. Use a normal approximation to find P(800 < x ≤ 820) .

Transportation officials tell us that 80% of drivers wear seat belts while driving. Find the probability that more than 576 drivers in a sample of 700 drivers wear seat belts.

Suppose that 88% of the stocks listed on a particular exchange increased in value yesterday. Let x be the number of stocks that increased in value yesterday in a random of 72 stocks listed on the exchange. Find the mean and standard deviation of x.

Determine the value of $e ^ { - a / \theta } \text { for } \theta = 2 \text { and } a = 3$

The probability density function for an exponential random variable x has a graph called a bell curve.

A certain baseball player hits a home run in 3% of his at-bats. Consider his at-bats as independent events. How many home runs do we expect the baseball player to hit in 600 at-bats?

Suppose that 67% of the employees of a company participate in the company's medical savings program. Let x be the number of employees who participate in the program in a random sample of 50 employees. Find the mean and standard deviation of x.

Quiz 5: Continuous Random Variables

Transportation officials tell us that 80% of drivers wear seat belts while driving. What is the probability of observing 633 or fewer drivers wearing seat belts in a sample of 850 drivers?

A certain baseball player hits a home run in 4% of his at-bats. Consider his at-bats as independent events. Find the probability that this baseball player hits at most 16 home runs in 650 at-bats?

A certain baseball player hits a home run in 6% of his at-bats. Consider his at-bats as independent events. Find the probability that this baseball player hits more than 32 home runs in 750 at-bats?

The exponential distribution is sometimes called the waiting-time distribution, because it is used to describe the length of time between occurrences of random events.

A loan officer has 66 loan applications to screen during the next week. If past record indicates that she turns down 17% of the applicants, what is the z-value associated with 61 or more of the 66 applications being rejected?

Transportation officials tell us that 80% of drivers wear seat belts while driving. What is the probability that between 538 and 546 drivers in a sample of 700 drivers wear seat belts?

Suppose that x has an exponential distribution with θ=1.5. Find P(x>1) \theta = 1.5 \text {. Find } P ( x > 1 ) θ=1.5. Find P(x>1)

A study of college students stated that 25% of all college students have at least one tattoo. In a random sample of 80 college students, let x be the number of the students that have at least one tattoo. Find the mean and standard deviation for this binomial distribution.

The exponential distribution has the property that its mean equals its standard deviation.

The exponential distribution is governed by two quantities, μ and σ, that determine its shape and location

Assume that x is a binomial random variable with n = 1000 and p = 0.80. Use a normal approximation to find P(800 < x ≤ 820) .

Transportation officials tell us that 80% of drivers wear seat belts while driving. Find the probability that more than 576 drivers in a sample of 700 drivers wear seat belts.

Suppose that 88% of the stocks listed on a particular exchange increased in value yesterday. Let x be the number of stocks that increased in value yesterday in a random of 72 stocks listed on the exchange. Find the mean and standard deviation of x.

Determine the value of e−a/θ for θ=2 and a=3e ^ { - a / \theta } \text { for } \theta = 2 \text { and } a = 3e−a/θ for θ=2 and a=3

The probability density function for an exponential random variable x has a graph called a bell curve.

A certain baseball player hits a home run in 3% of his at-bats. Consider his at-bats as independent events. How many home runs do we expect the baseball player to hit in 600 at-bats?

Suppose that 67% of the employees of a company participate in the company's medical savings program. Let x be the number of employees who participate in the program in a random sample of 50 employees. Find the mean and standard deviation of x.

Suppose that x has an exponential distribution with $\theta = 1.5 \text {. Find } P ( x > 1 )$

Determine the value of $e ^ { - a / \theta } \text { for } \theta = 2 \text { and } a = 3$