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Introduction To Probability And Statistics 14th Edition William Mendenhall, Robert Beaver, Barbara Beaver - Solutions

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. Find p(x). b. Construct a probability
A survey classified a large number of adults according to whether they were judged to need eyeglasses to correct their reading vision and whether they used eyeglasses when reading. The proportions falling into the four categories are shown in the table.If a single adult is selected from this large
A piece of electronic equipment contains six computer chips, two of which are defective. Three chips are selected at random, removed from the piece of equipment, and inspected. Let x equal the number of defectives observed, where x = 0, 1, or 2. Find the probability distribution for x. Express the
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. Graph p(x).
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 independent of the event that A wins any other, and the probability that A wins any one set is equal to
In Exercise 4.92 you found the probability distribution for x, the number of sets required to play a best-of-five-sets match, given that the probability that A wins any one set-call this P(A)-is .6. a. Find the expected number of sets required to complete the match for P(A) = .6. b. Find the
One professional golfer plays best on short-distance holes. Experience has shown that the numbers x of shots required for 3-, 4-, and 5-par holes have the probability distributions shown in the table:What is the golfer's expected score on these holes? a. A par-3 hole b. A par-4 hole c. A par-5 hole
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?
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 a profit. Suppose
Most coffee drinkers take a little time each day for their favorite beverage, and many take more than one coffee break every day. The table below, adapted from a Snapshot in USA Today, shows the probability distribution for x, the number of daily coffee breaks taken per day by coffee drinkers.a.
From experience, 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
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 determined
In his exciting novel Congo, Michael Crichton describes a search by Earth Resources Technology Service (ERTS), a geological survey company, for deposits of boron coated blue diamonds, diamonds that ERTS believes to be the key to a new generation of optical computers. In the novel, ERTS is racing
Consider a binomial random variable with n = 8 and p = .7. Let x be the number of successes in the sample.a. Find the probability that x is 3 or less.b. Find the probability that x is 3 or more.c. Find P(x < 3).d. Find P(x = 3).e. Find P(3 ≤ x ≤ 5).
Use Table 1 in Appendix I to find the sum of the binomial probabilities from x = 0 to x = k for these cases: a. n = 10, p = .1, k = 3 b. n = 15, p = .6, k = 7 c. n = 25, p = .5, k = 14
High gas prices may keep some American vacationers closer to home. However, when given a choice of getaway spots, 66% of U.S. leisure travelers indicated that they would like to visit national parks. A random sample of n = 100 leisure travelers is selected. a. What is the average of x, the number
Use Table 1 in Appendix I to evaluate the following probabilities for n = 6 and p = .8: a. P(x ≥ 4) b. P(x = 2) c. P(x < 2) d. P(x > 1) Verify these answers using the values of p(x) calculated in Exercise 5.7.
Find P(x ≤ k) for each of the following cases: a. n = 20, p = .05, k = 2 b. n = 15, p = .7, k = 8 c. n = 10, p = .9, k = 9
Use Table 1 in Appendix I to find the following: a. P(x < 12) for n = 20, p = .5 b. P(x ≤ 6) for n = 15, p = .4 c. P(x > 4) for n = 10, p = .4 d. P(x ≥ 6) for n = 15, p = .6 e. P(3 < x < 7) for n = 10, p = .5
Find the mean and standard deviation for a binomial distribution with these values: a. n = 1000, p = .3 b. n = 400, p = .01 c. n = 500, p = .5 d. n = 1600, p = .8
Find the mean and standard deviation for a binomial distribution with n = 100 and these values of p: a. p = .01 b. p = .9 c. p = .3 d. p = .7 e. p = .5
In Exercise 5.15, the mean and standard deviation for a binomial random variable were calculated for a fixed sample size, n = 100, and for different values of p. Graph the values of the standard deviation for the five values of p given in Exercise 5.15. For what value of p does the standard
Let x be a binomial random variable with n = 20 and p = .1.a. Calculate P(x ‰¤ 4) using the binomial formula.b. Calculate P(x ‰¤ 4) using Table 1 in Appendix I.c. Use the Excel output below to calculate P(x ‰¤ 4). Compare the results of parts a, b, and c.d. Calculate the mean and standard
Refer to Exercise 5.18. Assume that the sampling was conducted with replacement. That is, assume that the first ball was selected from the jar, observed, and then replaced, and that the balls were then mixed before the second ball was selected. Explain why x, the number of red balls observed, is or
Consider a binomial random variable with n = 9 and p = .3. Let x be the number of successes in the sample.a. Find the probability that x is exactly 2.b. Find the probability that x is less than 2.c. Find P(x > 2).d. Find P(2 ≤ x ≤ 4).
A meteorologist in Chicago recorded the number of days of rain during a 30-day period. If the random variable x is defined as the number of days of rain, does x have a binomial distribution? If not, why not? If so, are both values of n and p known?
A market research firm hires operators to conduct telephone surveys. The computer randomly dials a telephone number, and the operator asks the respondent whether or not he has time to answer some questions. Let x be the number of telephone calls made until the first respondent is willing to answer
In 2010, the average overall SAT score (Critical Reading, Math, and Writing) for college-bound students in the United States was 1509 out of 2400. Suppose that 45% of all high school graduates took this test and that 100 high school graduates are randomly selected from throughout the United
A home security system is designed to have a 99% reliability rate. Suppose that nine homes equipped with this system experience an attempted burglary. Find the probabilities of these events: a. At least one of the alarms is triggered. b. More than seven of the alarms are triggered. c. Eight or
In a certain population, 85% of the people have Rh-positive blood. Suppose that two people from this population get married. What is the probability that they are both Rh-negative, thus making it inevitable that their children will be Rh-negative?
Car color preferences change over the years and according to the particular model that the customer selects. In a recent year, suppose that 10% of all luxury cars sold were black. If 25 cars of that year and type are randomly selected, find the following probabilities: a. At least five cars are
The National Hockey League (NHL) has 80% of its players born outside the United States, and of those born outside the United States, 50% are born in Canada. Suppose that n = 12 NHL players were selected at random. Let x be the number of players in the sample who were born outside of the United
Records show that 30% of all patients admitted to a medical clinic fail to pay their bills and that eventually the bills are forgiven. Suppose n = 4 new patients represent a random selection from the large set of prospective patients served by the clinic. Find these probabilities: a. All the
Refer to Exercise 5.27 where 30% of all admitted patients fail to pay their bills and the debts are eventually forgiven. Suppose that the clinic treats 2000 different patients over a period of 1 year, and let x be the number of forgiven debts.a. What is the mean (expected) number of debts that have
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.a. What is the average number of fields sampled that are infested with whitefly?b. Within what limits would you
Evaluate these binomial probabilities: a. C82 (.3)2 (.7)6 b. C40 (.05)0 (.95)4 c. C103 (5)3 (5)7 d. C71 (.2)1 (.8)6
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 mouse chooses the red path. If
Six in 10 adults say lower back pain substantially limits their athletic activities. A random sample of n = 8 adults were asked if lower back pain was a limiting factor in their athletic activities. The printout below shows the cumulative and individual probabilities for a binomial random variable
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 n = 25 Americans who travel by car are asked how they determine where to stop for food and gas. Let x be the number in the sample who
The taste test for PTC (phenylthiocarbamide) is a favorite exercise for every human genetics class. It has been established that a single gene determines the characteristic, and that 70% of Americans are "tasters," while 30% are "non-tasters." Suppose that 20 Americans are randomly chosen and are
According to the Humane Society of the United States, there are approximately 77.5 million owned dogs in the United States, and approximately 40% of all U.S. households own at least one dog. Suppose that the 40% figure is correct and that 15 households are randomly selected for a pet ownership
Consider a Poisson random variable with µ = 2.5. Use the Poisson formula to calculate the following probabilities: a. P(x = 0) b. P(x = 1) c. P(x = 2) d. P(x ≤ 2)
Consider a Poisson random variable with µ = 3. Use the Poisson formula to calculate the following probabilities: a. P(x = 0) b. P(x = 1) c. P(x > 1)
Consider a Poisson random variable with µ = 3. Use Table 2 to find the following probabilities:a. P(x ≤ 3)b. P(x > 3)c. P(x = 3)d. P(3 ≤ x ≤ 5)
Consider a Poisson random variable with µ = 0.8. Use Table 2 to find the following probabilities:a. P(x = 0)b. P(x ≤ 2)c. P(x > 2)d. P(2 ≤ x ≤ 4)
Let x be a Poisson random variable with mean µ = 2. Calculate these probabilities: a. P(x = 0) b. P(x = 1) c. P(x > 1) d. P(x = 5)
Evaluate these binomial probabilities: a. C80 (.2)0 (.8)8 b. C81 (.2)1 (.8)7 c. C82 (.2)2 (.8)6 d. P(x ≤ 1) when n = 8, p = .2 e. P(two or fewer successes)
Let x be a Poisson random variable with mean µ = 2.5. Use Table 2 in Appendix I to calculate these probabilities: a. P(x = 5) b. P(x < 6) c. P(x = 2) d. P(1 ≤ x ≤ 4)
Let x be a binomial random variable with n = 20 and p = .1. a. Calculate P(x ≤ 2) using Table 1 in Appendix I to obtain the exact binomial probability. b. Use the Poisson approximation to calculate P(x ≤ 2). c. Compare the results of parts a and b. Is the approximation accurate?
To illustrate how well the Poisson probability distribution approximates the binomial probability distribution, calculate the Poisson approximate values for p(0) and p(1) for a binomial probability distribution with n = 25 and p = .05. Compare the answers with the exact values obtained from Table 1
The increased number of small commuter planes in major airports has heightened concern over air safety. An eastern airport has recorded a monthly average of five near misses on landings and takeoffs in the past 5 years. a. Find the probability that during a given month there are no near misses on
The number x of people entering the intensive care unit at a particular hospital on any one day has a Poisson probability distribution with mean equal to five persons per day. a. What is the probability that the number of people entering the intensive care unit on a particular day is two? Less than
According to a study conducted by the Department of Pediatrics at the University of California, San Francisco, children who are injured two or more times tend to sustain these injuries during a relatively limited time, usually 1 year or less. If the average number of injuries per year for
Refer to Exercise 5.45. a. Calculate the mean and standard deviation for x, the number of injuries per year sustained by a school-age child. b. Within what limits would you expect the number of injuries per year to fall? Exercise 5.45 According to a study conducted by the Department of Pediatrics
If a drop of water is placed on a slide and examined under a microscope, the number x of a specific type of bacteria present has been found to have a Poisson probability distribution. Suppose the maximum permissible count per water specimen for this type of bacteria is five. If the mean count for
An outbreak of E. coli infections in August of 2010 occurred in three Washington state day care centers. There were eight confirmed and six suspected cases of E. coli, with over 70 children awaiting test results.5 Outbreaks of E. coli infections for 2009 are reported to be less than 1 per 100,000,
Evaluate these probabilities: a. C21 C11 / C32 b. C40 C22 / C62 c. C22 C21 / C43
Let x be a binomial random variable with n = 7, p = .3. Find these values:a. P(x = 4)b. P(x ≤ 1)c. P(x > 1)d. µ = npe. σ = √npq
Let x be the number of successes observed in a sample of n = 4 items selected from a population of N = 8. Suppose that of the N = 8 items, 5 are considered "successes." a. Find the probability of observing all successes. b. Find the probability of observing one success. c. Find the probability of
Evaluate these probabilities: a. C31 C21 / C52 b. C42 C31 / C73 c. C54 C30 / C84
Let x be the number of successes observed in a sample of n = 5 items selected from N = 10. Suppose that, of the N = 10 items, 6 are considered "successes." a. Find the probability of observing no successes. b. Find the probability of observing at least two successes. c. Find the probability of
Let x be a hypergeometric random variable with N = 15, n = 3, and M = 4.a. Calculate p(0), p(1), p(2), and p(3).b. Construct the probability histogram for x.c. Use the formulas given in Section 5.4 to calculate µ = E(x) and σ2.d. What proportion of the population of measurements fall into the
A candy dish contains five blue and three red candies. A child reaches up and selects three candies without looking. a. What is the probability that there are two blue and one red candies in the selection? b. What is the probability that the candies are all red? c. What is the probability that the
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 chips. Compare your results with
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 p(x), the
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 teachers. Let us assume
Seeds are often treated with a fungicide for protection in poor-draining, wet environments. In a small-scale trial prior to a large-scale experiment to determine what dilution of the fungicide to apply, five treated seeds and five untreated seeds were planted in clay soil and the number of plants
Use the formula for the binomial probability distribution to calculate the values of p(x) and construct the probability histogram for x when n = 6 and p = .2.
Under what conditions can the Poisson random variable be used to approximate the probabilities associated with the binomial random variable? What application does the Poisson distribution have other than to estimate certain binomial probabilities?
Under what conditions would you use the hypergeometric probability distribution to evaluate the probability of x successes in n trials?
A balanced coin is tossed three times. Let x equal the number of heads observed.a. Use the formula for the binomial probability distribution to calculate the probabilities associated with x = 0, 1, 2, and 3.b. Construct the probability distribution.c. Find the mean and standard deviation of x,
Refer to Exercise 5.62. Suppose the coin is definitely unbalanced and the probability of a head is equal to p = .1. Follow the instructions in parts a, b, c, and d. Note that the probability distribution loses its symmetry and becomes skewed when p is not equal to 1/2.Exercise 5.62A balanced coin
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 survive for 10 years?
A city commissioner claims that 80% of all people in the city favor private garbage collection in contrast to collection by city employees. To check the 80% claim, you randomly sample 25 people and find that x, the number of people who support the commissioner's claim, is 22. a. What is the
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. What is the
Refer to Exercise 5.66. 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 you draw
Fewer Americans are really getting away while on vacation. In fact, among small business owners, more than half (51%) say they check in with the office at least once a day while on vacation; only 27% say they cut the cord completely. If 20 small business owners are randomly selected, and we assume
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 the 25 patients who have
Refer to Exercise 5.6. Construct the probability histogram for a binomial random variable x with n = 6 and p = .8. Use the results of Exercise 5.6; do not recalculate all the probabilities.Exercise 5.6Use the formula for the binomial probability distribution to calculate the values of p(x) and
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 n = 25 students produced 15 in favor of increased fees. What is the probability that 15 or fewer in the sample would favor the issue if student government
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 n = 200 students, recording x, the number of students age 30 or
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 the appropriate value of p,
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 the second package
One model for plant competition assumes that there is a zone of resource depletion around each plant seedling. Depending on the size of the zones and the density of the plants, the zones of resource depletion may overlap with those of other seedlings in the vicinity. When the seeds are randomly
A peony plant with red petals was crossed with another 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 ten seeds from this cross that were collected and germinated. a. Does the random
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 (bb) for this trait, 1/2
During the 1992 football season, the Los Angeles Rams (now the St. Louis Rams) had a bizarre streak of coin-toss losses. In fact, they lost the call 11 weeks in a row. a. The Rams' computer system manager said that the odds against losing 11 straight tosses are 2047 to 1. Is he correct? b. After
Insulin-dependent diabetes (IDD) among children occurs most frequently in persons of northern European descent. The incidence ranges from a low of 1-2 cases per 100,000 per year to a high of more than 40 per 100,000 in parts of Finland. Let us assume that an area in Europe has an incidence of 5
A new study by Square Trade indicates that smart-phones are 50% more likely to malfunction than simple phones over a 3-year period. Of smart-phone failures, 30% are related to internal components not working, and overall, there is a 31% chance of having your smart-phone fail over 3 years. Suppose
If x has a binomial distribution with p = .5, will the shape of the probability distribution be symmetric, skewed to the left, or skewed to the right?
Despite reports that dark chocolate is beneficial to the heart, 47% of adults still prefer milk chocolate to dark chocolate.11 Suppose a random sample of n = 5 adults is selected and asked whether they prefer milk chocolate to dark chocolate. a. What is the probability that all five adults say that
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 the wife is pregnant on
A Snapshot in USA Today shows that 60% of consumers say they have become more conservative spenders. When asked "What would you do first if you won $1 million tomorrow?" the answers had to do with somewhat conservative measures like "hire a financial advisor," or "pay off my credit card," or "pay
A procedure often used to control the quality of name-brand food products utilizes a panel of five "tasters." Each member of the panel tastes three samples, two of which are from batches of the product known to have the desired taste and the other from the latest batch. Each taster selects the
A public opinion research firm claims that approximately 70% of those sent questionnaires respond by returning the questionnaire. Twenty such questionnaires are sent out, and assume that the president's claim is correct. a. What is the probability that exactly 10 of the questionnaires are filled
Refer to Exercise 5.84. If n = 20 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 x = 10 of the 20 questionnaires are returned to the company, would you
A preliminary investigation reported that approximately 30% of locally grown poultry were infected with an intestinal parasite that, though not harmful to those consuming the poultry, decreased the usual weight growth rates in the birds. A diet supplement believed to be effective against this
In a food processing and packaging plant, there are, on the average, two packaging machine breakdowns per week. Assume 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 the probability that
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 find the
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 subject returns

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