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elementary statistics
Elementary Statistics 3rd Edition William Navidi, Barry Monk - Solutions
Let A and B be events. Now assume that P (A) > 0 and P (B) > 0.Is it possible for A and B to be both independent and mutually exclusive?Explain.Exercises 59–62 refer to the following situation:A medical test is available to determine whether a patient has a certain disease. To determine the
Explain why the following is not a probability distribution. x 6 7 8 9 10 P(x) 0.32 0.11 0.19 0.28 0.03
One of the most surprising probability distributions found in practice is given by a rule known as Benford’s law. This probability distribution concerns the first digits of numbers. The first digit of a number may be any of the digits 1, 2, 3, 4, 5, 6, 7, 8, or 9. It is reasonable to believe
Provide an example of a discrete random variable and explain why it is discrete.
Determine whether X is a binomial random variable.a. A fair die is rolled 20 times. Let X be the number of times the die comes up 6.b. A standard deck of 52 cards contains four aces. Four cards are dealt without replacement from this deck. Let X be the number that are aces.c. A simple random sample
A family has three children. If the genders of these children are listed in the order they are born, there are eight possible outcomes: BBB, BBG, BGB, BGG, GBB, GBG, GGB, and GGG. Assume these outcomes are equally likely. Let X represent the number of children that are girls. Find the probability
The number of traffic accidents at a certain intersection follows a Poisson distribution with λ = 2 per year. Let X be the number of accidents that occur in a 4-year period.a. Find P(7).b. Find P(Less than 4).c. Find P(More than 2).
Find the mean of the random variable X with the following probability distribution. x -2 1 4 5 P(x) 0.3 0.2 0.1 0.4
In a recent Pew poll, 50% of adults said that they play video games. Assume that 9 adults are randomly sampled.a. Use the binomial probability distribution to compute the probability that exactly four of them play video games.b. Use Table A.1 to find the probability that fewer than three of them
Provide an example of a continuous random variable and explain why it is continuous.
In a recent Pew poll, 10% of adults said that they consider themselves to be “gamers.”Assume that 18 adults are randomly sampled.a. Use the binomial probability distribution to compute the probability that exactly three of them consider themselves to be gamers.b. Use Table A.1 to find the
Refer to Problem 2.a. Find the variance of the random variable X.b. Find the standard deviation of the random variable X.Problem 2Find the mean of the random variable X with the following probability distribution. x -2 1 4 5 P(x) 0.3 0.2 0.1 0.4
Someone says that the following table shows the probability distribution for the number of boys in a family of four children. Is this possible? Explain why or why not. x 0 1 2 3 4 P(x) 0.12 0.37 0.45 0.25 0.18
The number of hits on a certain website follows a Poisson distribution with λ = 12 per minute. Let X be the number of hits that occur in a period of one-half minute.a. Find P(5).b. Find P(Less than 2).c. Find P(More than 3).
Which of the following tables represent probability distributions? a. x P(x) b. x P(x) C. x P(x) 0 0.45 4 0.27 1 0.02 1 0.15 5 0.15 2 0.41 2 0.30 6 0.11 3 0.24 3 0.10 7 0.34 4 0.33 8 0.25
When a population mean is unknown, people will often approximate it with the mean of a large sample. Explain why this is justified.
An environmental scientist is counting the number of bacteria in a sample of wastewater. The number of bacteria has a Poisson distribution with λ = 3 per microliter. What is the probability that exactly 9 bacteria are found in a volume of t = 4 microliters?
Find the missing value that makes the following a valid probability distribution. x 2 3 5 8 10 P(x) 0.23 0.12 0.09 ? 0.37
Gregor Mendel discovered the basic laws of heredity by studying pea plants. In one experiment, he produced plants whose parent plants contained genes for both green and yellow pods. Mendel’s theory states that the offspring of two such parents has probability 0.75 of having green pods. Assume
During a recent academic year, approximately 1.7 million students took one or more AP tests. Following is the frequency distribution of the number of AP tests taken by students who took one or more AP tests.Let X represent the number of exams taken by a student who took one or more.a. Construct the
Following is the probability distribution of a random variable that represents the number of extracurricular activities a college freshman participates in.a. Find the probability that a student participates in exactly two activities.b. Find the probability that a student participates in more than
In the Georgia Cash 3 Lottery game, your probability of winning is 0.001. The game is played twice daily, so 730 games are played in a year. If you play twice a day for a year, what is the probability that you win exactly twice?
The following table presents a probability distribution for the number of pets each family has in a certain neighborhood.Construct a probability histogram. Number of pets Probability 012 3. 4 0.4 0.2 0.2 0.1 0.1
Provide an example of a binomial random variable and explain how each condition for the binomial distribution is fulfilled.
In a binomial distribution, there are _______________ possible outcomes for each trial.In Exercises 5–7, fill in each blank with the appropriate word or phrase.
There are 5000 undergraduates registered at a certain college. Of them, 478 are taking one course, 645 are taking two courses, 568 are taking three courses, 1864 are taking four courses, 1357 are taking five courses, and 88 are taking six courses. Let X be the number of courses taken by a student
The Poisson distribution is used to describe events that occur in ______________ or _____________ .In Exercises 5 and 6, fill in each blank with the appropriate word or phrase.
Refer to Problem 5.Find the probability that a randomly selected family has:a. 1 or 2 petsb. More than 2 petsc. No more than 3 petsd. At least 1 petProblem 5The following table presents a probability distribution for the number of pets each family has in a certain neighborhood.Construct a
Following is a probability histogram for the number of children a woman has. The numbers on the tops of the rectangles are the heights.a. What is the probability that a randomly chosen woman has exactly two children?b. What is the probability that a randomly chosen woman has fewer than two
To compute a binomial probability, we must know both the success probability and the number n of ___________________ .In Exercises 5–7, fill in each blank with the appropriate word or phrase.
Refer to Problem 5.Find the mean number of pets.Problem 5The following table presents a probability distribution for the number of pets each family has in a certain neighborhood.Construct a probability histogram. Number of pets Probability 012 3. 4 0.4 0.2 0.2 0.1 0.1
The mean of the Poisson random variable X with rate λ and interval length t is __________________ .In Exercises 5 and 6, fill in each blank with the appropriate word or phrase.
Provide an example of a Poisson random variable and explain how each condition for the Poisson distribution is fulfilled.
If X is a binomial random variable with n trials and success probability p, the standard deviation of X isσX = __________________ .In Exercises 5–7, fill in each blank with the appropriate word or phrase.
Refer to Problem 5.Find the standard deviation of the number of pets.Problem 5The following table presents a probability distribution for the number of pets each family has in a certain neighborhood.Construct a probability histogram. Number of pets Probability 012 3. 4 0.4 0.2 0.2 0.1 0.1
A true–false quiz with 10 questions was given to a statistics class. Following is the probability distribution for the score of a randomly chosen student. Find the mean score and interpret the result. x 5 6 7 8 9 10 P(x) 0.04 0.16 0.36 0.24 0.12 0.08
For a Poisson random variable with rate λ and time interval t, the possible values of x are 0, 1, 2, ..., λ.
Following is the probability distribution for the age of a student at a certain public high school.a. Find the variance of the ages.b. Find the standard deviation of the ages. x 13 14 15 16 17 18 P(x) 0.08 0.24 0.23 0.28 0.14 0.03
The trials in a binomial distribution are independent.In Exercises 8–10, determine whether the statement is true or false. If the statement is false, rewrite it as a true statement.
A binomial random variable with n trials can sometimes have a value greater than n.In Exercises 8–10, determine whether the statement is true or false. If the statement is false, rewrite it as a true statement.
In a Poisson distribution, the mean and variance are equal.
At a cell phone battery plant, 5% of cell phone batteries produced are defective. A quality control engineer randomly collects a sample of 50 batteries from a large shipment from this plant and inspects them for defects. Find the probability thata. None of the batteries are defective.b. At least
A numerical outcome of a probability experiment is called a ____________.In Exercises 9–12, fill in each blank with the appropriate word or phrase.
λ = 2, t = 5, P(5)In Exercises 9–18, determine the indicated probability for a Poisson random variable with the given values of λ and t.
Refer to Problem 9.Find the mean and standard deviation for the number of defective batteries in the sample of size 50.Problem 9At a cell phone battery plant, 5% of cell phone batteries produced are defective. A quality control engineer randomly collects a sample of 50 batteries from a large
Let X be the number of days during the next month that it rains. Does X have a binomial distribution?Why or why not?
The mean of a binomial random variable is found by multiplying the number of trials by the success probability.In Exercises 8–10, determine whether the statement is true or false. If the statement is false, rewrite it as a true statement.
The sum of all the probabilities in a discrete probability distribution must be equal to _____________ .In Exercises 9–12, fill in each blank with the appropriate word or phrase.
λ = 0.5, t = 4, P(3)In Exercises 9–18, determine the indicated probability for a Poisson random variable with the given values of λ and t.
A meteorologist states that the probability of rain tomorrow is 0.4 and the probability of rain on the next day is 0.6.Assuming these probabilities are accurate, and that the rain events are independent, find the probability distribution for X, the number of days out of the next two that it rains.
In a college with 5000 students, 100 are randomly chosen to complete a survey in which they rate the quality of the cafeteria food. Let X be the number of freshmen who are chosen. Does X have a binomial distribution?Why or why not?
Ten students are chosen from a statistics class of 25 students.Let X be the number who got an A in the class.In Exercises 11–16, determine whether the random variable X has a binomial distribution. If it does, state the number of trials n. If it does not, explain why not.
λ = 0.1, t = 10, P(2)In Exercises 9–18, determine the indicated probability for a Poisson random variable with the given values of λ and t.
The number of large packages delivered by a courier service follows a Poisson distribution with a rate of 5 per day. Find the probability thata. Exactly 4 large packages are delivered on a given day.b. Fewer than 6 large packages are delivered over a 2-day period.c. At least one large package is
Ten students are chosen from a statistics class of 300 students.Let X be the number who got an A in the class.In Exercises 11–16, determine whether the random variable X has a binomial distribution. If it does, state the number of trials n. If it does not, explain why not.
As the sample size increases, the sample mean approaches the _______________ mean.In Exercises 9–12, fill in each blank with the appropriate word or phrase.
λ = 1, t = 2, P(0)In Exercises 9–18, determine the indicated probability for a Poisson random variable with the given values of λ and t.
The number of text messages received on a certain person’s cell phone follows a Poisson distribution with the rate of 10 messages per hour. What is the mean number of messages received in an 8-hour period?
Refer to Exercise 12.Find the mean number of particles that are withdrawn.Exercise 12The concentration of particles in a suspension is 2 per milliliter. A volume of 3 milliliters is withdrawn.Find the following probabilities.
A coin is tossed seven times. Let X be the number of heads obtained.In Exercises 11–16, determine whether the random variable X has a binomial distribution. If it does, state the number of trials n. If it does not, explain why not.
To find the mean of a discrete random variable, multiply each possible value of the random variable by its probability, then add the products.In Exercises 13–16, determine whether the statement is true or false. If the statement is false, rewrite it as a true statement.
λ = 0.2, t = 10, P(At least one)In Exercises 9–18, determine the indicated probability for a Poisson random variable with the given values of λ and t.
Refer to Problem 13.What are the variance and standard deviation of the number of messages received in an 8-hour period?Problem 13The number of text messages received on a certain person’s cell phone follows a Poisson distribution with the rate of 10 messages per hour. What is the mean number of
Refer to Exercise 12.Find the standard deviation of the number of particles that are withdrawn.Exercise 12The concentration of particles in a suspension is 2 per milliliter. A volume of 3 milliliters is withdrawn.Find the following probabilities.
A die is tossed three times. Let X be the sum of the three numbers obtained.In Exercises 11–16, determine whether the random variable X has a binomial distribution. If it does, state the number of trials n. If it does not, explain why not.
The expected value is the mean amount gained or lost.In Exercises 13–16, determine whether the statement is true or false. If the statement is false, rewrite it as a true statement.
λ = 0.3, t = 8, P(At least one)In Exercises 9–18, determine the indicated probability for a Poisson random variable with the given values of λ and t.
Huntington’s disease is a genetically transmitted disease that causes degeneration of nerve cells in the brain. The probability that a person carries a gene for Huntington’s disease is 0.00005 (i.e., 1∕20,000). In a sample of 100,000 people, what is the probability that exactly four people
A coin is tossed until a head appears. Let X be the number of tosses.In Exercises 11–16, determine whether the random variable X has a binomial distribution. If it does, state the number of trials n. If it does not, explain why not.
The possible values of a discrete random variable cannot be listed.In Exercises 13–16, determine whether the statement is true or false. If the statement is false, rewrite it as a true statement.
λ = 1, t = 4, P(No more than 5)In Exercises 9–18, determine the indicated probability for a Poisson random variable with the given values of λ and t.
A random sample of 250 voters is chosen from a list of 10,000 registered voters. Let X be the number who support the incumbent mayor for reelection.In Exercises 11–16, determine whether the random variable X has a binomial distribution. If it does, state the number of trials n. If it does not,
The standard deviation is found by squaring the variance.In Exercises 13–16, determine whether the statement is true or false. If the statement is false, rewrite it as a true statement.
λ = 0.2, t = 7, P(More than 2)In Exercises 9–18, determine the indicated probability for a Poisson random variable with the given values of λ and t.
n = 5, p = 0.7, P(3)In Exercises 17–26, determine the indicated probability for a binomial experiment with the given number of trials n and the given success probability p. Then find the mean, variance, and standard deviation.
The number of heads in 100 tosses of a coin In Exercises 17–26, determine whether the random variable described is discrete or continuous.
n = 10, p = 0.2, P(1)In Exercises 17–26, determine the indicated probability for a binomial experiment with the given number of trials n and the given success probability p. Then find the mean, variance, and standard deviation.
The number of people in line at the bank at a randomly chosen time In Exercises 17–26, determine whether the random variable described is discrete or continuous.
λ = 0.3, t = 5, P(Fewer than 3)In Exercises 9–18, determine the indicated probability for a Poisson random variable with the given values of λ and t.
n = 20, p = 0.6, P(8)In Exercises 17–26, determine the indicated probability for a binomial experiment with the given number of trials n and the given success probability p. Then find the mean, variance, and standard deviation.
The weight of a randomly chosen student’s backpack In Exercises 17–26, determine whether the random variable described is discrete or continuous.
In a certain city, the number of potholes on a major street has a Poisson distribution with a rate of 3 per mile. Let X represent the number of potholes in a 2-mile stretch of road.Finda. P(4)b. P(More than 1)c. P(Between 5 and 7 inclusive)d. μXe. σX
n = 14, p = 0.3, P(8)In Exercises 17–26, determine the indicated probability for a binomial experiment with the given number of trials n and the given success probability p. Then find the mean, variance, and standard deviation.
The amount of rain during the next thunderstorm In Exercises 17–26, determine whether the random variable described is discrete or continuous.
The number of flaws in a given area of aluminum foil follows a Poisson distribution with a mean of 3 per square meter. Let X represent the number of flaws in a 1-square-meter sample of foil.a. P(5)b. P(0)c. P(Less than 2)d. P(Greater than 1)e. μXf. σX
n = 3, p = 0.4, P(0)In Exercises 17–26, determine the indicated probability for a binomial experiment with the given number of trials n and the given success probability p. Then find the mean, variance, and standard deviation.
The number of children absent from school on a randomly chosen day In Exercises 17–26, determine whether the random variable described is discrete or continuous.
The number of flaws in a certain type of lumber has a Poisson distribution with a rate of 0.5 per meter of length.a. What is the probability that a board 3 meters in length has no flaws?b. What is the probability that a board 4 meters in length has at least one flaw?
n = 6, p = 0.8, P(6)In Exercises 17–26, determine the indicated probability for a binomial experiment with the given number of trials n and the given success probability p. Then find the mean, variance, and standard deviation.
The time it takes to drive to the airport In Exercises 17–26, determine whether the random variable described is discrete or continuous.
The number of times that screen flicker appears in video recorded on a certain type of digital video recorder (DVR) has a Poisson distribution with a rate of 0.2 per minute.a. What is the probability that there are no screen flickers in 8 minutes of video?b. What is the probability that there are
n = 8, p = 0.2, P(Fewer than 3)In Exercises 17–26, determine the indicated probability for a binomial experiment with the given number of trials n and the given success probability p. Then find the mean, variance, and standard deviation.
The final exam score of a randomly chosen student from last semester’s statistics class In Exercises 17–26, determine whether the random variable described is discrete or continuous.
The number of tweets received by a certain Twitter user is a Poisson random variable with a mean rate of 8 tweets per hour.a. What is the probability that 5 tweets are received in a given hour?b. What is the probability that 10 tweets are received in 1.5 hours?c. What is the probability that fewer
n = 15, p = 0.9, P(14 or more)In Exercises 17–26, determine the indicated probability for a binomial experiment with the given number of trials n and the given success probability p. Then find the mean, variance, and standard deviation.
The amount of time you wait at a bus stop In Exercises 17–26, determine whether the random variable described is discrete or continuous.
Grandma bakes chocolate chip cookies in batches of 100.She puts 600 chips into each batch, so that the number of chips in a randomly chosen cookie has a Poisson distribution with λ = 6 chips per cookie.a. Grandma gives you one cookie. What is the probability that it contains exactly eight chips?b.
n = 50, p = 0.03, P(2 or fewer)In Exercises 17–26, determine the indicated probability for a binomial experiment with the given number of trials n and the given success probability p. Then find the mean, variance, and standard deviation.
The height of a randomly chosen college student In Exercises 17–26, determine whether the random variable described is discrete or continuous.
The number of trees of a certain species in a certain forest has a Poisson distribution with mean of 10 trees per acre.a. What is the probability that there will be exactly 18 trees in a 2-acre region?b. Find the mean number of trees in a 2-acre region.c. Find the standard deviation of the number
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