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Questions and Answers of Statistics

Show that the failure-rate function is given by if and only if the time to failure distribution is the Weibull distribution 0. " alt="Show that the failure-rate function is given by if and"
According to a study published by a group of sociologists at the University of Massachusetts, approximately 49% of the Valium users in the state of Massachusetts are white-collar workers. What is the
The exponential distribution is frequently applied to the waiting times between successes in a Poisson process. If the number of calls received per hour by a telephone answering service is a Poisson
When a is a positive integer n, the gamma distribution is also known as the Erlang distribution. Setting a = n in the gamma distribution on page 195, the Erlang distribution is It can be shown that
A manufacturer of a certain type of large machine wishes to buy rivets from one of two manufacturers. It is important that the breaking strength of each rivet exceed 10,000 psi. Two manufacturers (A
According to a recent census, almost 65% of all households in the United States were composed of only one or two persons. Assuming that this percentage is still valid today, what is the probability
The life of a certain type of device has an advertised failure rate of 0.01 per hour. The failure rate is constant and the exponential distribution applies.(a) What is the mean time to failure?(b)
In a chemical processing plant it is important that the yield of a certain type of batch product stay above 80%. If it stays below 80% for an extended period of time, the company loses money.
Consider an electrical component failure rate of once every 5 hours. It is important to consider the time that it takes for 2 components to fail.(a) Assuming that the gamma distribution applies, what
The elongation of a steel bar under a particular load has been established to be normally distributed with a mean of 0.05 inch and a = 0.01 inch. Find the probability that the elongation is(a) Above
A controlled satellite is known to have an error (distance from target) that is normally distributed with mean zero and standard deviation 4 feet. The manufacturer of the satellite defines a
A technician plans to test a certain type of resin developed in the laboratory to determine the nature of the time it takes before bonding takes place. It is known that the mean time to bonding is 3
Consider the information in Review Exercise 6.64. What is the probability that less than 200 hours will elapse before 2 failures occur?
For Review Exercise 6.70 what is the mean and variance of the time that elapses before 2 failures occur?
The average rate of water usage (thousands of gallons per hour) by a certain community is known to involve the lognormal distribution with parameters p = 5 and a = 2. It is important for planning
For Review Exercise 6.72, what is the mean of the average water usage per hour in thousands of gallons?
In Exercise 6.52 on page 206, the lifetime of a transistor is assumed to have a gamma distribution with mean 10 weeks and standard deviation V50 weeks. Suppose that the gamma distribution assumption
Consider Exercise 6.50 on page 206. The beta distribution has considerable application in reliability problems in which the basic random variable is a proportion in the practical scenario illustrated
Consider now Review Exercise 3.76. The density function of the time Z in minutes between calls to an electrical supply store is given by (a) What is the mean time between calls? (b) What is the
Consider Review Exercise 6.76. Given the assumption of the exponential distribution, what is the mean number of calls per hour? What is the variance in the number of calls per hour?
In a human factor experimental project, it has been determined that the reaction time of a pilot to a visual stimulus is normally distributed with a mean of 1/2 second and standard deviation of 2/5
The length of time between breakdowns of an essential piece of equipment is important in the decision of the use of auxiliary equipment. An engineer thinks that the best "model" for time between
The length of life, in hours, of a drill bit in a mechanical operation has a Weibull distribution with a = 2 and 0 = 50. Find the probability that the bit will fail before 10 hours of usage.
Derive the cdf for the Weibull distribution.
In Review Exercise 6.80, explain why the nature of the scenario would likely not lend itself to the exponential distribution.
Form the relationship between the chi-squared random variable and the gamma random variable, prove that the mean of the chi-squared random variable is v and the variance is 2v.
The length of time, in seconds, that a computer user reads his (or her) e-mail is distributed as a lognormal random variable with p = 1.8 and a2 = 4.0.(a) What is the probability that the user reads
An employee is selected from a staff of 10 to supervise a certain project by selecting a tag at random from a box containing 10 tags numbered from 1 to 10. Find the formula for the probability
Twelve people: are given two identical speakers to listen to for differences, if any. Suppose that these people answered by guessing only. Find the probability that three people claim to have heard a
Find the mean and variance of the random variable A' of Exercise 5.1.
In a certain city district the need for money to buy drugs is stated as the: reason for 75% of all thefts. Find the probability that among the next 5 theft cases reported in this district,(a) Exactly
According to Chemical Engineering Progress (Nov. 1990), approximately 30% of all pipe work failures in chemical plants are caused by operator error.(a) What is the probability that out of the next 20
According to a survey by the Administrative Management Society, one-half of U.S. companies give employees 4 weeks of vacation after they have been with the company for 15 years. Find the probability
One prominent physician claims that 70% of those with lung cancer are chain smokers. If his assertion is correct,(a) Find the: probability that of 10 such patients recently admitted to a hospital,
According to a study published by a group of University of Massachusetts sociologists, approximately 60% of the Valium users in the state of Massachusetts first took Valium for psychological
In testing a certain kind of truck tire over a rugged terrain, it is found that 25% of the trucks fail to complete the test run without a blowout. Of the next 15 trucks tested, find the probability
A nationwide survey of seniors by the University of Michigan reveals that almost 70% disapprove of daily pot smoking, according to a report in Parade. If 12 seniors are selected at random and asked
The probability that a patient recovers from a delicate heart operation is 0.9. What is the probability that exactly 5 of the next 7 patients having this operation survive?
A traffic control engineer reports that. 75% of the vehicles passing through a checkpoint are from within the state. What is the probability that fewer than 4 of the next 9 vehicles arc from out of
A study examined national attitudes about antidepressants. The study revealed that approximately 70% believe "antidepressants do not really cure anything they just cover up the real trouble."
It is known that the percentage of wins for the Chicago Bulls basketball team going into the playoffs for the 1996-97 season was 87.7. Round the 87.7 to 90 in order to use Table A.1(a) What is the
It is known that 60% of mice inoculated with a serum are protected from a certain disease. If 5 mice are inoculated, find the probability that(a) None contracts the disease;(b) Fewer than 2 contract
Suppose that airplane engines operate independently and fail with probability equal to 0.4. Assuming that a plane makes a safe flight if at least one-half of its engines run, determine whether a
If X represents the number of people in Exercise 5.13 who believe that antidepressants do not cure but only cover up the real problem, find the mean and variance of X when 5 people are selected at
(a) In Exercise 5.9, how many of the 15 trucks would you expect to have blowouts?(b) According to Chebyshev's theorem, there is a probability of at least 3/4 that the number of trucks among the next
As a student drives to school, he encounters a traffic signal. This traffic signal stays green for 35 seconds, yellow for 5 seconds, and red for 60 seconds. Assume that the student goes to school
According to USA Today (March 18, 1997), of 4 million workers in the general workforce, 0.8% tested positive for drugs. Of those testing positive, 22.5% were cocaine users and 54.4% marijuana
The surface of a circular dart board has a small center circle called the bull's-eye and 20 pie-shaped regions numbered from 1 to 20. Each of the pie-shaped regions is further divided into three
According to a genetics theory, a certain cross of guinea pigs will result in red, black, and white offspring in the ratio 8:4:4. Find the probability that among 8 offspring 5 will be red, 2 black,
The probabilities are 0.4, 0.2, 0.3, and 0.1, respectively, that a delegate to a certain convention arrived by air, bus. Auto mobile, or train what is the probability that among 9 delegates randomly
A safety engineer claims that only 40% of all workers wear safety helmets when they eat lunch at the workplace. Assuming that his claim is right, find the probability that 4 of 6 workers randomly
Suppose that for a very large shipment of integrated-circuit chips, the probability of failure for any one chip is 0.10. Assuming that the assumptions underlying the binomial distributions are met,
Assuming that 6 in 10 automobile accidents are due mainly to a speed violation find the probability that, among 8 automobile accidents 6 will be due mainly to a speed violation(a) By using the
If the probability that a fluorescent light has a useful life of at least 800 hours is 0.9, find the probabilities that among 20 such lights(a) Exactly 18 will have a useful life of at least 800
A manufacturer knows that on the average 20% of the electric toasters which lie makes will require repairs within 1 year after they are sold. When 20 toasters are randomly selected, find appropriate
If 7 cards are dealt from an ordinary deck of 52 playing cards, what is the probability that(a) Exactly 2 of them will be face cards?(b) At least 1 of them will be a queen?
To avoid detection at customs, a traveler places 6 narcotic tablets in a bottle containing 9 vitamin pills that are similar in appearance. If the customs official selects 3 of the tablets at random
A homeowner plants 6 bulbs selected at random from a box containing 5 tulip bulbs and 4 daffodil bulbs. What is the probability that he planted 2 daffodil bulbs and 4 tulip bulbs?
From a lot of 10 missiles, 4 are selected at random and fired. If the lot contains 3 defective missiles that will not fire, what is the probability that(a) All 4 will fire?(b) At most 2 will not fire?
A random committee of size 3 is selected from 4 doctors and 2 nurses. Write a formula for the probability distribution of the random variable X representing the number of doctors on the committee.
What is the probability that a waitress will refuse to serve alcoholic beverages to only 2 minors if she randomly checks the IDs of 5 students from among 9 students of which 4 arc not of legal age?
A company is interested in evaluating its current inspection procedure on shipments of 50 identical items. The procedure is to take a sample of 5 and pass the shipment if no more than 2 are found to
A manufacturing company uses an acceptance scheme on production items before they are shipped. The plan is a two-stage one. Boxes of 25 are readied for shipment and a sample of 3 is tested for
Suppose that the manufacturing company of Exercise 5.36 decides to change its acceptance scheme. Under the new scheme an inspector takes one at random, inspects it, and then replaces it in the box; a
In Exercise 5.32, how many defective missiles might we expect to be included among the 4 that are selected? Use Chebyshev's theorem to describe the variability of the number of defective missiles
If a person is dealt 13 cards from an ordinary deck of 52 playing cards several times, how many hearts per hand can he expect? Between what two values would you expect the number of hearts to fall at
It is estimated that 4000 of the 10,000 voting residents of a town are against a new sales tax. If 15 eligible voters are selected at random and asked their opinion, what is the probability that at
An annexation suit is being considered against a county subdivision of 1200 residences by a neighboring city. If the occupants of half the residences object to being annexed, what is the probability
Among 150 IRS employees in a large city, only 30 are women. If 10 of the applicants are chosen at random to provide free: fax assistance for the residents of this city, use the binomial approximation
A nationwide survey of 17,000 seniors by the University of Michigan revealed that almost 70% disapprove of daily pot smoking. If 18 of these seniors are selected at random and asked their opinions,
Find the probability of being dealt a bridge hand of 13 cards containing 5 spades, 2 hearts 3 diamonds and 3 clubs.
A foreign student club lists as its members 2 Canadians, 3 Japanese, 5 Italians, and 2 Germans. If a committee of 4 is selected at random, find the probability that(a) All nationalities are
An urn contains 3 green balls, 2 blue balls, and 4 red balls. In a random sample: of 5 balls, find the probability that both blue balls and at least 1 reel ball are selected.
Population studies of biology and the environment often tag and release subjects in order to estimate size and degree: of certain features in the population. Ten animals of a certain population
A large company has an inspection system for the batches of small compressors purchased from vendors. A batch typically contains 15 compressors. In the inspection system a random sample of 5 is
A government task force suspects that some manufacturing companies are in violation of federal pollution regulations with regard to dumping a certain type of product. Twenty firms are under suspicion
Every hour, 10,000 cans of soda are filled by a machine, among which 300 under filled cans are produced. Each hour a sample of 30 cans is randomly selected and the number of ounces of soda per can is
The probability that a person, living in a certain city, owns a dog is estimated to be 0.3. Find the probability that the tenth person randomly interviewed in that city is the fifth one to own a dog.
A scientist inoculates several mice, one at a time, with a disease germ until he finds 2 that have contracted the disease. If the probability of contracting the disease is 1/6, what is the
An inventory study determines that, on average, demands for a particular item at a warehouse are made 5 times per day. What is the probability that on a given day this item is requested?(a) More than
Find the probability that a person flipping a coin gets(a) The third head on the seventh flip;(b) The first head on the fourth flip.
Three people toss a fair coin and the odd man pays for coffee. If the coins all turn up the same, they are tossed again. Find the probability that fewer than 4 tosses are needed.
According to a study published by a group of University of Massachusetts sociologists, about two thirds of the 20 million persons in this country who take Valium are women. Assuming this figure to be
The probability that a student pilot passes the written test for a private pilot's license is 0.7. Find the probability that the student will pass the test(a) On the third try;(b) Before the fourth
On average a certain intersection results in 3 traffic accidents per month. What is the probability that for any given month at this intersection(a) Exactly 5 accidents will occur?(b) Less than 3
A secretary makes 2 errors per page, on average. What is the probability that on the next page he or she will make(a) 4 or more errors?(b) No errors?
A certain area of the eastern United States is, on average, hit by 6 hurricanes a year. Find the probability that for a given year that area will be hit by(a) Fewer than 4 hurricanes;(b) Anywhere
Suppose the probability is 0.8 that any given person will believe a tale about the transgressions of a famous actress. What is the probability that(a) The sixth person to hear this tale is the fourth
The average number of field mice per acre in a 5-acre wheat field is estimated to be 12. Find the probability that fewer than 7 field mice are found(a) On a given acre;(b) On 2 of the next 3 acres
A restaurant chef prepares a tossed salad containing, on average, 5 vegetables. Find the probability that the salad contains more than 5 vegetables(a) On a given day;(b) On 3 of the next 4 days;(c)
The probability that a person will die from a certain respiratory infection is 0.002. Find the probability that fewer than 5 of the next 2000 so infected will die.
Suppose that, on average, 1 person in 1000 makes a numerical error in preparing his or her income tax return. If 10,000 forms are selected at random and examined, find the probability that 6, 7, or 8
The probability that a student fails the screening test for scoliosis (curvature of the spine) at a local high school is known to be 0.004. Of the next 1875 students who are screened for scoliosis,
(a) Find the mean and variance in Exercise 5.64 of the random variable X representing the number of persons among 2000 that die from the respiratory infection.(b) According to Chebyshev's theorem,
(a) Find the mean and variance in Exercise 5.65 of the random variable X representing the number of persons among 10,000 who make an error in preparing their income tax returns.(b) According to
An automobile manufacturer is concerned about a fault in the braking mechanism of a particular model. The fault can, on rare occasions, cause a catastrophe at high speed. The distribution of the
Changes in airport procedures require considerable planning. Arrival rates of aircraft are important factors that must be taken into account. Suppose small aircraft arrive at a certain airport,
The number of customers arriving per hour at a certain automobile service facility is assumed to follow a Poisson distribution with mean A = 7.(a) Compute the probability that more than 10 customers
Consider Exercise 5.66. What is the mean number of students who fail the test?
The probability that a person dies when he contracts a virus infection is 0.001. Of the next 4000 so infected, what is the mean number that will die?

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