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Introduction To Probability And Statistics 15th Edition William Mendenhall Iii , Robert Beaver , Barbara Beaver - Solutions
19. Dark Chocolate Despite reports that dark chocolate is beneficial to the heart, 47% of adults still prefer milk chocolate to dark chocolate.14 Suppose a random sample of n55 adults is selected and asked whether they prefer milk chocolate to dark chocolate.a. What is the probability that all five
18. Problems with Your New Smartphone? According to TECH.CO, there are seven most common smartphone issues and there are simple fixes for most of them. One of the problems are phone crashes which accounted for approximately 25% of the performance issues.13 Suppose that smartphones are shipped in
17. Diabetes in Children Diabetes incidence rates in the United States have skyrocketed in kids and teens over the last 15 years. Type I or insulin dependent diabetes now has an incidence rate of 21.7 cases per 100,000 while the incidence rate for Type II (adult onset diabetes), which is associated
16. Dominant Traits The alleles for black (B) and white (b) feather color in chickens show incomplete dominance; individuals with the gene pair Bb have“blue” feathers. When one individual that is homozygous dominant (BB) for this trait is mated with an individual that is homozygous recessive
15. Plant Density One model for plant competition assumes that there is a zone of resource depletion around each plant seedling. When the seeds are randomly dispersed over a wide area, the number of neighbors that a seedling may have usually follows a Poisson distribution with a mean equal to the
14. What’s for Breakfast? A packaging experiment is conducted by placing two different package designs for a breakfast food side by side on a supermarket shelf.On a given day, 25 customers purchased a package of the breakfast food from the supermarket. Let x equal the number of buyers who choose
23. Questionnaires, continued Refer to Exercise 22.If n520 questionnaires are sent out,a. What is the average number of questionnaires that will be returned?b. What is the standard deviation of the number of questionnaires that will be returned?c. If x510 of the 20 questionnaires are returned to
24. Poultry Problems A preliminary investigation reported that approximately 30% of locally grown poultry were infected with an intestinal parasite that decreased the usual weight growth rates in the birds.A diet supplement believed to be effective against this parasite was added to the birds’
1. If the 52 reported cases represented a rate 50% higher than the statewide rate, what is a reasonable estimate of m, the average number of such cancer cases statewide? How safe is it to live near a nuclear reactor? Men who lived in a coastal strip that extends 32 kilometers north from a nuclear
32. Fast Food and Gas Stations Forty percent of all Americans who travel by car look for gas stations and food outlets that are close to or visible from the highway.Suppose a random sample of n525 Americans who travel by car are asked how they determine where to stop for food and gas. Let x be the
31. Color Preferences in Mice In a psychology experiment, the researcher designs a maze in which a mouse must choose one of two paths, colored either red or blue, at each of 10 intersections. At the end of the maze, the mouse is given a food reward. The researcher counts the number of times the
30. Whitefly Infestation Suppose that 10% of the fields in a given agricultural area are infested with the sweet potato whitefly. One hundred fields in this area are randomly selected and checked for whitefly. Within what limits would you expect to find the number of infested fields, with
29. Need College? Is college necessary? About 50% of Californians think that it is not, citing mounting costs and large student debts.15 A random sample of 25 Californians is selected and assume that the p5.5 figure is correct. Let x be the number who believe that college is not important.a. What
28. Eating on the Run How do you survive when there’s no time to eat—fast food, no food, a protein bar, candy? A USA Today snapshot indicates that 36%of women aged 25–55 say that, when they are too busy to eat, they get fast food from a drive-thru. A random sample of 100 women aged 25–55 is
27. Enrolling in College A West Coast university has found that about 90% of its accepted applicants for enrollment in the freshman class will actually enroll. In 2020, 1360 applicants were accepted to the university.Within what limits would you expect to find the size of the freshman class at this
26. Safe Drivers? Evidence shows that the probability that a driver will be involved in a serious automobile accident during a given year is .01. A particular corporation employs 100 full-time traveling sales reps.Based on this evidence, use the Poisson approximation to the binomial distribution to
25. Machine Breakdowns In a food processing and packaging plant, there are, on the average, two packaging machine breakdowns per week. Assume that the weekly machine breakdowns follow a Poisson distribution.a. What is the probability that there are no machine breakdowns in a given week?b. Calculate
13. Probability of Rain To check the accuracy of a particular weather forecaster, records were checked only for those days when the forecaster predicted rain “with 30% probability.” A check of 25 of those days indicated that it rained on 10 of the 25.a. If the forecaster is accurate, what is
12. Gray Hair on Campus College campuses are graying!According to a recent article, one in four college students is aged 30 or older. Assume that the 25% figure is accurate, that your college is representative of colleges at large, and that you sample n5200 students, recording x, the number of
11. Student Fees A student government states that 80% of all students favor an increase in student fees to subsidize a new recreational area. A random sample of n525 students produced 15 in favor of increased fees.What is the probability that 15 or fewer in the sample would favor the issue if
34. Seed Treatments Seeds are often treated with a fungicide for protection in poor-draining, wet environments.In a small-scale trial, five treated seeds and five untreated seeds were planted in clay soil and the number of plants emerging from the treated and untreated seeds were recorded. Suppose
33. Teaching Credentials In southern California, a growing number of persons pursuing a teaching credential are choosing paid internships over traditional student teaching programs. A group of eight candidates for three teaching positions consisted of five paid interns and three traditional student
32. Unbiased Choices A company has five applicants for two positions: two women and three men. Suppose that the five applicants are equally qualified and that no preference is given for choosing either gender. Let x equal the number of women chosen to fill the two positions.a. Write the formula for
31. Defective Computer Chips A piece of electronic equipment contains six computer chips, two of which are defective. Three computer chips are randomly chosen for inspection, and the number of defective chips is recorded. Find the probability distribution for x, the number of defective computer
30. Cramming A student prepares for an exam by studying a list of 10 problems. She can solve 6 of them.For the exam, the instructor selects 5 problems at random from the list of 10. What is the probability that the student can solve all 5 problems on the exam?
29. Voter Registration A city ward consists of 200 registered voters of whom 125 are registered Republicans and 75 are registered with other parties. On voting day, n510 people are selected at random for an exit poll in this ward.a. What is the probability distribution, p(x), for x, the number of
Draw three cards randomly from a standard deck of 52 cards and let x be the number of kings in the draw. Evaluate the probabilities 28. Would the probability distribution in Exercise 27 change if x were defined to be the number of aces in the draw?
Draw three cards randomly from a standard deck of 52 cards and let x be the number of kings in the draw. Evaluate the probabilities 27. p(x) for x=0, 1, 2, 3
Draw three cards randomly from a standard deck of 52 cards and let x be the number of kings in the draw. Evaluate the probabilities 26. P(x=3)
35. Bad Wiring Improperly wired control panels were mistakenly installed on two of eight large automated machine tools. It is uncertain which of the machine tools have the defective panels, and a sample of four tools is randomly chosen for inspection. What is the probability that the sample will
1. Tennis, Anyone? Two tennis professionals, A and B, are scheduled to play a match; the winner is the first player to win three sets in a total that cannot exceed five sets. The event that A wins any one set is P(A) 5 .6, and is independent of the event that A wins any other set. Let x equal the
10. Psychosomatic Problems A psychiatrist believes that 80% of all people who visit doctors have problems of a psychosomatic nature. She decides to select 25 patients at random to test her theory.a. Assuming that the psychiatrist’s theory is true, what is the expected value of x, the number of
9. Checking In A new study shows more than half of workers ages 55 and over do not check in at all while on vacation, while a whopping 62% of Millennial workers ages 18 to 34 check in with the office at least once or twice a week during their vacation.11 If n520 Millennials are randomly selected
8. Integers II Refer to Exercise 7. Twenty people are asked to select a number from 0 to 9. Eight of them choose a 4, 5, or 6.a. If the choice of any one number is as likely as any other, what is the probability of observing eight or more choices of the numbers 4, 5, or 6?b. What conclusions would
7. Integers If a person is given the choice of an integer from 0 to 9, is it more likely that he or she will choose an integer near the middle of the sequence than one at either end?a. If the integers are equally likely to be chosen, find the probability distribution for x, the number chosen.b.
6. Garbage Collection To check a claim that 80% of all people in the city favor private garbage collection in contrast to collection by city employees, you randomly sample 25 people and find that x, the number of people who support the claim, is 22.a. What is the probability of observing at least
5. A Color Recognition Experiment An experiment is run as follows—the colors red, yellow, and blue are each flashed on a screen for a short period of time. A subject views the colors and is asked to choose the one he feels was flashed for the longest time. The experiment is repeated three times
4. Accident Insurance Accident records collected by an automobile insurance company give the following information: The probability that an insured driver has an automobile accident is .15; if an accident has occurred, the damage to the vehicle amounts to 20% of its market value with probability
3. The PGA One professional golfer plays best on short-distance holes. Experience has shown that the numbers x of shots required for par-3, par-4, and par-5 holes have the probability distributions shown in the table:What is the golfer’s expected score on these holes?a. A par -3 holeb. A par -4
2. Heavy Equipment A heavy-equipment salesman can contact either one or two customers per day with probabilities 1/3 and 2/3, respectively. Each contact will result in either no sale or a $50,000 sale with probabilities 9/10 and 1/10, respectively. What is the expected value of his daily sales?
Draw five cards randomly from a standard deck of 52 cards, and let x be the number of red cards in the draw. Evaluate the probabilities 25. P(x1)
Let x be a binomial random variable with n=7 and p=.5. Find the values of the quantities 15. =npq
Consider a binomial random variable with n=8 and p=.7. Let x be the number of successes in the sample. Evaluate the probabilities 3. P(x3)
Consider a binomial random variable with n=8 and p=.7. Let x be the number of successes in the sample. Evaluate the probabilities 2. P(x3)
1. List the five identifying characteristics of the binomial experiment.
A regimen consisting of a daily dose of vitamin C was tested to see if it might be effective in preventing the common cold. Ten people followed the prescribed regimen for a year, and eight survived the winter without a cold. Suppose that, without vitamin C, the probability of surviving the winter
Refer to Example 5.9 and the binomial random variable x with n55 and p5.6. Use the cumulative binomial table to find the remaining binomial probabilities, p(0), p(1), p(2), p(4), and p(5). Construct the probability histogram for the random variable x and describe its shape and location.
Use the cumulative binomial table for n55 and p5.6 to find the probabilities of these events:1. Exactly three successes 2. Three or more successes
At any given time, a professional basketball player can make a free throw with probability equal to .8. Suppose he shoots four free throws.1. What is the probability that he will make exactly two free throws?2. What is the probability that he will make at least one free throw?
Find P(x52) for a binomial random variable with n510 and p5.1.
A patient fills a prescription for a medication to be taken twice a day for 10 days. Unknown to the pharmacist and the patient, the 20 tablets consist of 18 pills of the prescribed medication and 2 pills that are its generic equivalent. The patient selects two pills at random for the first day’s
Consider a binomial random variable with n=8 and p=.7. Let x be the number of successes in the sample. Evaluate the probabilities 4. P(x
Consider a binomial random variable with n=8 and p=.7. Let x be the number of successes in the sample. Evaluate the probabilities 5. P(x=3)
Let x be a binomial random variable with n=7 and p=.5. Find the values of the quantities 14. = np
Let x be a binomial random variable with n=7 and p=.5. Find the values of the quantities 13. P(x>1)
Let x be a binomial random variable with n=7 and p=.5. Find the values of the quantities 12. P(x1)
Let x be a binomial random variable with n=7 and p=.5. Find the values of the quantities 11. P(x=4)
Consider a binomial random variable with n = 9 and p = .3. Let x be the number of successes in the sample. Evaluate the probabilities 10. P(2x4)
Consider a binomial random variable with n = 9 and p = .3. Let x be the number of successes in the sample. Evaluate the probabilities 9. P(x>2)
Consider a binomial random variable with n = 9 and p = .3. Let x be the number of successes in the sample. Evaluate the probabilities 8. The probability that x is less than 2.
Consider a binomial random variable with n = 9 and p = .3. Let x be the number of successes in the sample. Evaluate the probabilities 7. The probability that x is exactly 2.
Consider a binomial random variable with n=8 and p=.7. Let x be the number of successes in the sample. Evaluate the probabilities 6. P(3x5)
Suppose there are approximately 1,000,000 adults in a county and an unknown proportion p favors term limits for politicians. A sample of 1000 adults will be chosen in such a way that every one of the 1,000,000 adults has an equal chance of being selected, and each adult is asked whether he or she
38. Orchestra Politics The board of directors of a major symphony orchestra has voted to create a committee to handle employee complaints. The committee will consist of the president and vice president of the symphony board and two orchestra representatives. The two orchestra representatives will
37. Actuaries A CEO is considering buying an insurance policy to cover possible losses incurred by marketing a new product. If the product is a complete failure, a loss of $800,000 would be incurred; if it is only moderately successful, a loss of $250,000 would be incurred. Insurance actuaries have
25. Unbiased Choices Five applicants have applied for two positions: two women and three men. All are equally qualified and there is no preference for choosing either gender. Let x be the number of women chosen to fill the two positions. find and graph the probability distribution for x. Then
24. RU Texting? Of adults 18 years and older, 47%admit to texting while driving.1 Three adults are randomly selected and x, the number who admit to texting while driving is recorded. find and graph the probability distribution for x. Then calculate the mean, variance, and standard deviation.
23. A Pair of Dice Toss a pair of dice and record x, the sum of the numbers on the two upper faces. find and graph the probability distribution for x. Then calculate the mean, variance, and standard deviation.
22. Dice Let x be the number observed on the throw of a single balanced die. find and graph the probability distribution for x. Then calculate the mean, variance, and standard deviation.
21. What is the probability that x is 3 or less?Use the probability distribution for the random variable x to answer x 0 1 2 3 4 p(x) .1 .3 .3 ? .1
20. What is the probability that x is greater than 2?Use the probability distribution for the random variable x to answer x 0 1 2 3 4 p(x) .1 .3 .3 ? .1
19. Calculate the population mean, variance, and standard deviation.Use the probability distribution for the random variable x to answer x 0 1 2 3 4 p(x) .1 .3 .3 ? .1
18. Construct a probability histogram for p(x).Use the probability distribution for the random variable x to answer x 0 1 2 3 4 p(x) .1 .3 .3 ? .1
17. Find p(3).Use the probability distribution for the random variable x to answer x 0 1 2 3 4 p(x) .1 .3 .3 ? .1
26. Defective Equipment A piece of electronic equipment contains 6 computer chips, two of which are defective. Three chips are randomly selected and inspected, and x, the number of defective chips in the selection is recorded. find and graph the probability distribution for x. Then calculate the
27. Grocery Visits Let x represent the number of times a customer visits a grocery store in a 1-week period.Assume this is the probability distribution of x:Find the expected value of x, the average number of times a customer visits the store. x p(x) | 0 12 3 .1 .4 A .1
36. Shipping Charges A shipping company knows that the cost of delivering a small package within 24 hours is$14.80. The company charges $15.50 for shipment but guarantees to refund the charge if delivery is not made within 24 hours. If the company fails to deliver only 2%of its packages within the
35. Coffee Breaks Most coffee drinkers take a little time each day for their favorite beverage, and many take more than one coffee break every day. The following table, adapted from a USA Today snapshot, shows the probability distribution for x, the number of coffee breaks taken per day by coffee
34. FDA Testing The maximum patent life for a new drug is 17 years. Subtracting the length of time required by the FDA for testing and approval of the drug provides the actual patent life of the drug—that is, the length of time that a company has to recover research and development costs and make
33. Insuring Your Diamonds You can insure a $50,000 diamond for its total value by paying a premium of D dollars. If the probability of loss in a given year is estimated to be .01, what premium should the insurance company charge if it wants the expected gain to equal$1000?
32. Fire Alarms A fire-detection device uses three temperature-sensitive cells acting independently of one another so that any one or more can activate the alarm.Each cell has a probability p5.8 of activating the alarm when the temperature reaches 578C or higher. Let x equal the number of cells
31. Roulette A roulette wheel contains 38 pockets—the numbers 1 through 36, 0, and 00. The wheel is spun and the “winning” pocket is recorded, with any one pocket just as likely as any other. Suppose you bet $5 on the number 18. The payoff on this type of bet is usually$35 for a $1 bet. What
30. Fire Insurance In a county containing a large number of rural homes, 60% of the homes are insured against fire. Four rural homeowners are chosen at random from this county, and x are found to be insured against fire. Find the probability distribution for x.What is the probability that at least
29. Drilling Oil Wells Past experience has shown that, on the average, only 1 in 10 wells drilled hits oil. Let x be the number of drillings until the first success (oil is struck). Assume that the drillings represent independent events.a. Find p(1), p(2), and p(3).b. Give a formula for p(x).c.
28. Which Key Fits? A key ring contains four office keys that are identical in appearance, but only one will open your office door. Suppose you randomly select one key and try it. If it does not fit, you randomly select one of the three remaining keys. If that key does not fit, you randomly select
16. If you were to select a very large number of values of x from the population, would most fall into the interval m 62s? Explain.Use the probability distribution for the random variable x to answer X 0 1 2 3 4 5 p(x).1 .3 .4 .1 ? .05
Let x be a Poisson random variable, find the probabilities for x using the Poisson formula. 4. 3; P(x = 0), P(x-1), and P(x>1).
48. Tay–Sachs Disease Tay–Sachs disease is a genetic disorder that is usually fatal in young children. If both parents are carriers of the disease, the probability that their offspring will develop the disease is approximately.25. Suppose a husband and wife are both carriers of the disease and
47. Plant Genetics A peony plant with red petals was crossed with a peony plant having streaky petals. The probability that an offspring from this cross has red flowers is .75. Let x be the number of plants with red petals resulting from 10 seeds from this cross that were collected and
46. Football Coin Tosses During the 1992 football season, the Los Angeles Rams had a bizarre streak of cointoss losses. In fact, they lost the call 11 weeks in a row.4a. The Rams’ computer system manager said that the odds against losing 11 straight tosses are 2047 to 1. Is he correct?b. After
45. Earthquakes! Suppose that 1 out of every 10 homeowners in the state of California has invested in earthquake insurance. If 15 homeowners are randomly chosen to be interviewed,a. What is the probability that at least one had earthquake insurance?b. What is the probability that four or more have
44. McDonald’s or Burger King? Suppose that 50%of all young adults prefer McDonald’s to Burger King when asked to state a preference. A group of 10 young adults were randomly selected and their preferences recorded.a. What is the probability that more than 6 preferred McDonald’s?b. What is
43. Engine Failure Suppose the four engines of a commercial aircraft are arranged to operate independently and that the probability of in-flight failure of a single engine is .01. What is the probability of the following events on a given flight?a. No failures are observed.b. No more than one
42. Successful Surgeries A new surgical procedure is said to be successful 80% of the time. Suppose the operation is performed five times and the results are assumed to be independent of one another.a. What is the probability that all five operations are successful.b. What is the probability that
41. Cancer Survivor Rates The 10-year survival rate for bladder cancer is approximately 50%. If 20 people who have bladder cancer are properly treated for the disease, what is the probability that:a. At least 1 will survive for 10 years?b. At least 10 will survive for 10 years?c. At least 15 will
40. Coins, continued Refer to Exercise 39. Suppose the coin is definitely unbalanced and the probability of a head is equal to p5.1. Follow the instructions in partsa,b, c, andd. Notice that the probability distribution loses its symmetry and becomes skewed when p is not equal to 1/2.
49. Stressed Out A subject is taught to do a task in two different ways. Studies have shown that when subjected to mental strain and asked to perform the task, the subject most often reverts to the method first learned, regardless of whether it was easier or more difficult.If the probability that a
50. Blood Types In a certain population, 85% of the people have Rh-positive blood. Suppose that two people from this population marry. What is the probability that they are both Rh-negative, thus making it inevitable that their children will be Rh-negative?
Let x be a Poisson random variable, find the probabilities for x using the Poisson formula. 3. u 2.5; P(x=0), P(x=1), P(x=2), and P(x2).
2. What application does the Poisson distribution have other than to estimate certain binomial probabilities?
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