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inferential statistics
Probability And Statistics For Engineers And Scientists 9th Global Edition Ronald E. Walpole, Raymond Myers, Sharon L. Myers, Keying E. Ye - Solutions
6.53 In a biomedical research study, it was determined that the survival time, in weeks, of an animal subjected to a certain exposure of gamma radiation has a gamma distribution with α = 5 and β = 10.(a) What is the mean survival time of a randomly selected animal of the type used in the
6.52 Derive the mean and variance of the Weibull distribution.
6.51 The lives of a certain automobile seal have the Weibull distribution with failure rate Z(t) =1/√t.Find the probability that such a seal is still intact after 4 years.
6.50 If the proportion of a brand of television requiring service during the first year of operation is a random variable having a beta distribution with α = 3 and β = 2, what is the probability that at least 80% of the new models of this brand sold this year will require service during their
6.49 Suppose the random variable X follows a beta distribution with α = 1 and β = 3.(a) Determine the mean and median of X.(b) Determine the variance of X.(c) Find the probability that X > 1/3.
6.48 Derive the mean and variance of the beta distribution.
6.47 Suppose that the service life, in years, of a hearing aid battery is a random variable having a Weibull distribution with α = 1/2 and β = 2.(a) How long can such a battery be expected to last?(b) What is the probability that such a battery will be operating after 2 years?
6.46 The life of a street bulb follows an exponential distribution, with an average life β = 3 years. The bulbs are replaced whenever they fail. Out of the 1000 street bulbs installed in a city, what is the probability that at most 250 of them will need to be replaced during the first year?
6.45 At a train reservation counter, one man completes his reservation with a mean time of 3 minutes.Service completion time is assumed to follow exponential distribution. Out of the 5 customers in queue, what is the probability that at least 4 will complete their reservation within 3 minutes?
6.44 The water supply board of a metropolitan city reveals that the each family consumes an average of 20 liters of drinking water per day, with a standard deviation of√200 liters. Let X denote the drinking water consumption per family and follow the gamma distribution.(a) Find α and β.(b) Find
6.43 (a) Find the mean and variance of the daily water consumption in Exercise 6.40.(b) According to Chebyshev’s theorem, there is a probability of at least 3/4 that the water consumption on any given day will fall within what interval?
6.42 Suppose that the time, in hours, required to service a motorbike is a random variable X having a gamma distribution, with α = 2 and β = 12. What is the probability that on the next service call,(a) at most 2 hours of service will be required?(b) at least 1 hour of service will be required?
6.41 If a random variable X has a gamma distribution, with α = 2, β = 1, find P(1.6 > X > 2.8).
6.40 In a certain city, the daily consumption of water(in millions of liters) follows approximately a gamma distribution with α = 2 and β = 3. If the daily capacity of that city is 9 million liters of water, what is the probability that on any given day the water supply is inadequate?
6.39 Use the gamma function with y =√2x to show that Γ(1/2) =√π.
6.38 A telemarketing company has a special letteropening machine that opens and removes the contents of an envelope. If the envelope is fed improperly into the machine, the contents of the envelope may not be removed or may be damaged. In this case, the machine is said to have “failed.”(a) If
6.37 The systolic blood pressure X of 30-year-old men is approximately normally distributed with a mean of 123 mmHg and standard deviation of 6 mmHg.(a) Find the probability that the blood pressure of a randomly selected 30-year-old man exceeds 127 mmHg.(b) Out of 250 randomly selected men of 30
6.36 A common practice of airline companies is to sell more tickets for a particular flight than there are seats on the plane, because customers who buy tickets do not always show up for the flight. Suppose that the percentage of no-shows at flight time is 2%. For a particular flight with 197
6.35 A firm assembling electronics has a record of 98% perfect assemblies. They export the assembled items in a lot of 50 units.(a) What is the probability that a lot will contain more than 2 defectives?(b) What is the probability that a lot will contain at most one defective?
6.34 A pair of dice is rolled 360 times. What is the probability of obtaining a sum of 12(a) at least 5 times?(b) between 12 and 18 times, both included?(c) exactly 6 times?
6.33 Statistics released by the National Highway Traffic Safety Administration and the National Safety Council show that on an average weekend night, 1 out of every 10 drivers on the road is drunk. If 400 drivers are randomly checked next Saturday night, what is the probability that the number of
6.32 A pin manufacturing company knows that the chance of producing a defective item is 4%. The company markets the items with a promise that in a pack of 250, no more than 10 pins will be defective. What are the chances of the company keeping up its claim?Use the normal approximation to the
6.31 One-sixth of the male freshmen entering a large state school are out-of-state students. If the students are assigned at random to dormitories, 180 to a building, what is the probability that in a given dormitory at least one-fifth of the students are from out of state?
6.30 A village administrator claims that, on average, 75% of the villagers are literate. The authorities decided to verify the claim by testing 100 villagers selected at random. They decided to accept the claim if 70% or more of the villages were found to be literate.(a) What is the probability
6.29 If 20% of the residents in a U.S. city prefer a white cell phone over any other color available, what is the probability that among the next 1000 cell phone purchased in that city(a) between 170 and 185 inclusive will be white?(b) at least 210 but not more than 225 will be white?
6.28 As part of the research on “the role of English as a gateway to knowledge”, a survey is conducted among 1000 college students, in which 72% of the students agree with the statement. If 100 students are picked at random, what is the probability that(a) at least 80 of them agree with the
6.27 In a city, 4% of the adolescents are alcoholic.Out of the 100 adolescents randomly selected, what is the probability that(a) between 8 and 18 of them are alcoholics?(b) fewer than 5 are alcoholics?
6.26 A certain batch contains 5% defectives defective.If 100 apples are randomly examined, what is the probability that the number of defective apples(a) exceeds 15?(b) is less than 10?
6.25 In a textile manufacturing company, 2% of the items are known to be defective. The quality control team decides to select 200 items produced by the company for examination. If none of the units are found to be defective, the process continues. Use the normal approximation to the binomial to
6.24 The probability of success of 600 students appearing in a test independently is 0.40. Using normal approximation, find the probability that(a) between 215 and 250 students pass, both inclusive;(b) exactly 270 students pass;(c) fewer than 205 and more than 260 students pass.
6.23 The marks for 1000 applicants in a college admission test are approximately normally distributed, with a mean of 125 and standard deviation of 15. If the college decided to consider the applicants who scored at least 100, how many of the students will be rejected?Note that marks are recorded
6.22 If a set of observations is normally distributed, what percent of these differ from the mean by,(a) More than 1.50σ;(b) Less than 0.75σ.
6.21 The tensile strength of a certain metal component is normally distributed with a mean of 10,000 kilograms per square centimeter and a standard deviation of 100 kilograms per square centimeter. Measurements are recorded to the nearest 50 kilograms per square centimeter.(a) What proportion of
6.20 The weights of a large number of miniature poodles are approximately normally distributed with a mean of 8 kilograms and a standard deviation of 0.9 kilogram. If measurements are recorded to the nearest tenth of a kilogram, find the fraction of these poodles with weights(a) over 9.5
6.19 A company spends an average of $18.60 per hour on power, with a standard deviation of $1.80. If the power charges are approximately normally distributed and paid to the nearest cent, (a) what percentage of the operating hours will have an expense of less than $16.50 per hour?(b) what will be
6.18 The weights of 1000 students are normally distributed with a mean of 62.5 kilograms and a standard deviation of 2.7 kilograms. Assuming that the weights are recorded to the nearest half-kilograms, how many of these students would you expect to have weights(a) less than 55 kilograms?(b) between
6.17 An engineering firm produces machines with an average life of 8 years and standard deviation of 2 years.The firm wishes to introduce a warranty scheme in which it would like to replace all the dysfunctional machines under warranty with new ones. But they do not wish to do so for more than 5%
6.16 In the November 1990 issue of Chemical Engineering Progress, a study discussed the percent purity of oxygen from a certain supplier. Assume that the mean was 99.61 with a standard deviation of 0.08. Assume that the distribution of percent purity was approximately normal.(a) What percentage of
6.15 A lawyer commutes daily from his suburban home to his midtown office. The average time for a one-way trip is 24 minutes, with a standard deviation of 3.8 minutes. Assume the distribution of trip times to be normally distributed.(a) What is the probability that a trip will take at least 1/2
6.14 The inside diameter of the washers produced by a certain company is normally distributed with a mean of 0.60 centimeters and standard deviation of 0.004 centimeters.(a) What percentage of the washers have an inside diameter of more than 0.61 centimeters?(b) Obtain the probability that the
6.13 A group of individuals with standard health conditions are put on an experimental diet for one month.The gain in weights of these individuals after a month is normally distributed. They average 1450 grams, with a standard deviation of 250 grams. Find the probability that the gain in weight for
6.12 The average length of steel nails is 5 centimeters, with a standard deviation of 0.05 centimeters.Assuming that the lengths are normally distributed, what percentage of the nails are(a) longer than 5.05 centimeters?(b) between 4.95 and 5.05 centimeters in length?(c) shorter than 4.90
6.11 A soft-drink machine is regulated so that it discharges an average of 200 milliliters per cup. If the amount of drink is normally distributed with a standard deviation equal to 15 milliliters,(a) what fraction of the cups will contain more than 224 milliliters?(b) what is the probability that
6.10 According to Chebyshev’s theorem, the probability that any random variable assumes a value within 2 standard deviations of the mean is at least 0.75. If it is known that the probability distribution of a random variable X is normal, with mean μ and variance σ2 , what is the exact value of
6.9 Given the normally distributed variable X with a mean of 20 and standard deviation of 2, find(a) P(X < 16);(b) the value of k such that P(X < k) = 0.4090;(c) the value of k such that P(X > k) = 0.8599;(d) P(17 < X < 22).
6.8 Given a normal distribution with μ = 30 andσ = 6, find(a) the normal curve area to the right of x = 17;(b) the normal curve area to the left of x = 22;(c) the normal curve area between x = 32 and x = 41;(d) the value of x that has 80% of the normal curve area to the left;(e) the two values of
6.7 Given the standard normal distribution, find the value of k such that(a) P(Z > k) = 0.9625;(b) P(Z < k) = 0.6255;(c) P(0.17 < Z < k) = 0.367.
6.6 Find the value of z if the area under a standard normal curve(a) to the right of z is 0.3745;(b) to the left of z is 0.3050;(c) between –z and 0, with –z < 0, is 0.4838;(d) between –z and z with z > 0, is 0.90.
6.5 Given a standard normal distribution, find the area under the curve that lies(a) to the left of z = −1.10;(b) to the right of z = 1.645;(c) between z = −2.43 and −0.45;(d) to the left of z = 0.45;(e) to the right of z = −0.45;(f) between z = −0.45 and z = 0.45.
6.4 A train arrives at a station every 15 minutes. It is assumed that the waiting time for a particular individual is a random variable with a continuous uniform distribution.(a) What is the probability that the individual waits more than 10 minutes?(b) What is the probability that the individual
6.3 The daily milk consumption, in liters, dispensed by a milk booth located in a city is a random variable,X, having a continuous uniform distribution with A = 90 and B = 120. Find the probability that on a given day, the amount of milk dispensed by this booth will be (a) at most 100 liters;(b)
6.2 Suppose X follows a continuous uniform distribution from 0 to 5. Determine the conditional probability, P(X < 3.5/X ≥ 1).
6.1 Given a continuous uniform distribution, show that(a) μ = A+B 2 and(b) σ2 = (B−A)2 12 .
5.102 Group Project: Divide the class into two groups of approximately equal size. The students in group 1 will each toss a coin 10 times (n1 ) and count the number of heads obtained. The students in group 2 will each toss a coin 40 times (n2 ) and again count the number of heads. The students in
5.101 The manufacturer of a tricycle for children has received complaints about defective brakes in the product.According to the design of the product and considerable preliminary testing, it had been determined that the probability of the kind of defect in the complaint was 1 in 10,000 (i.e.,
5.100 There are two vacancies in a certain university’s statistics department. Six individuals apply. Two have expertise in linear models, and one has expertise in applied probability. The search committee is instructed to choose the two applicants randomly.(a) What is the probability that the
5.99 Go back to Review Exercise 5.95(a). Recompute the probability using the binomial distribution.Comment.
5.98 Suppose it is important that the overall missile defense system be as near perfect as possible.(a) Assuming the quality of the screens is as indicated in Review Exercise 5.97, how many are needed to ensure that the probability that a missile gets through undetected is 0.0001?(b) Suppose it is
5.97 National security requires that defense technology be able to detect incoming projectiles or missiles.To make the defense system successful, multiple radar screens are required. Suppose that three independent screens are to be operated and the probability that any one screen will detect an
5.96 Consider the situation of Review Exercise 5.95.It has been determined that the sampling plan should be extensive enough that there is a high probability, say 0.9, that if as many as 2 defectives exist in the lot of 50 being sampled, at least 1 will be found in the sampling. With these
5.95 A production process outputs items in lots of 50.Sampling plans exist in which lots are pulled aside periodically and exposed to a certain type of inspection.It is usually assumed that the proportion defective is very small. It is important to the company that lots containing defectives be a
5.94 A production process produces electronic component parts. It is presumed that the probability of a defective part is 0.01. During a test of this presumption, 500 parts are sampled randomly and 15 defectives are observed.(a) What is your response to the presumption that the process is 1%
5.93 It is known by researchers that 2 in 100 people carry a gene that leads to the inheritance of a certain chronic disease. In a random sample of 1000 individuals, what is the probability that fewer than 11 individuals carry the gene? Use a Poisson approximation.Again, using the approximation,
5.92 A couple decides to continue to have children until they have two males. Assuming that P(male) = 0.5, what is the probability that their second male is their fifth child?
5.91 Consider the information in Review Exercise 5.90. The drilling company feels that it will “hit it big” if the second success occurs on or before the sixth attempt. What is the probability that the driller will hit it big?
5.90 An oil drilling company ventures into various locations, and its success or failure is independent from one location to another. Suppose the probability of a success at any specific location is 0.25.(a) What is the probability that the driller drills at 10 locations and has 1 success?(b) The
5.89 The acceptance scheme for purchasing lots containing a large number of batteries is to test no more than 75 randomly selected batteries and to reject a lot if a single battery fails. Suppose the probability of a failure is 0.001.(a) What is the probability that a lot is accepted?(b) What is
5.88 The potential buyer of a particular engine requires(among other things) that the engine start successfully 10 consecutive times. Suppose the probability of a successful start is 0.98. Let us assume that the outcomes of attempted starts are independent.(a) What is the probability that the
5.87 Imperfections in computer circuit boards and computer chips lend themselves to statistical treatment.For a particular type of board, the probability of a diode failure is 0.02 and the board contains 200 diodes.(a) What is the mean number of failures among the diodes?(b) What is the
5.86 Suppose that out of 500 lottery tickets sold, 100 pay off at least the cost of the ticket. Now suppose that you buy 5 tickets. Find the probability that you will win back at least the cost of 3 tickets.
5.85 (a) Suppose that you throw 3 dice. Find the probability that you get at least one 2.(b) Suppose that you throw 2 dice 20 times. Find the probability that you get at least one (1, 1), that is, snake-eyes.
5.84 A local drugstore owner knows that, on average, 100 people enter his store each hour.(a) Find the probability that in a given 3-minute period nobody enters the store.(b) Find the probability that in a given 3-minute period more than 5 people enter the store.
5.83 A company generally purchases large lots of a certain kind of electronic device. A method is used that rejects a lot if 2 or more defective units are found in a random sample of 100 units.(a) What is the probability of rejecting a lot that is 2%defective?(b) What is the probability of
5.82 An electronic switching device occasionally malfunctions, but the device is considered satisfactory if it makes, on average, no more than 0.20 error per hour.A particular 5-hour period is chosen for testing the device.If no more than 1 error occurs during the time period, the device will be
5.81 An electronics firm claims that the proportion of defective units from a certain process is 5%. A buyer has a standard procedure of inspecting 15 units selected randomly from a large lot. On a particular occasion, the buyer found 5 items defective.(a) What is the probability of this
5.80 Service calls come to a maintenance center according to a Poisson process, and on average, 2.7 calls are received per minute. Find the probability that (a) no more than 4 calls come in any minute;(b) fewer than 2 calls come in any minute;(c) more than 10 calls come in a 5-minute period.
5.79 A car rental agency at a local airport has available 5 Fords, 7 Chevrolets, 4 Dodges, 3 Hondas, and 4 Toyotas. If the agency randomly selects 9 of these cars to chauffeur delegates from the airport to the downtown convention center, find the probability that 2 Fords, 3 Chevrolets, 1 Dodge, 1
5.78 An automatic welding machine is being considered for use in a production process. It will be considered for purchase if it is successful on 99% of its welds. Otherwise, it will not be considered efficient.A test is to be conducted with a prototype that is to perform 100 welds. The machine will
5.77 During a manufacturing process, 15 units are randomly selected each day from the production line to check the percent defective. From historical information it is known that the probability of a defective unit is 0.05. Any time 2 or more defectives are found in the sample of 15, the process is
5.76 The refusal rate for telephone polls is known to be approximately 20%. A newspaper report indicates that 50 people were interviewed before the first refusal.(a) Comment on the validity of the report. Use a probability in your argument.(b) What is the expected number of people interviewed
5.75 Computer technology has produced an environment in which robots operate with the use of microprocessors.The probability that a robot fails during any 6-hour shift is 0.10. What is the probability that a robot will operate through at most 5 shifts before it fails?
5.74 It is known that 2% of people whose luggage is screened at an airport have questionable objects in their luggage. What is the probability that a string of 15 people pass through screening successfully before an individual is caught with a questionable object? What is the expected number of
5.73 Hospital administrators in large cities anguish about traffic in emergency rooms. At a particular hospital in a large city, the staff on hand cannot accommodate the patient traffic if there are more than 10 emergency cases in a given hour. It is assumed that patient arrival follows a Poisson
5.72 Potholes on a highway can be a serious problem, and are in constant need of repair. With a particular type of terrain and make of concrete, past experience suggests that there are, on the average, 2 potholes per mile after a certain amount of usage. It is assumed that the Poisson process
5.71 For a certain type of copper wire, it is known that, on average, 1.2 flaws occur per millimeter. Assuming that the number of flaws is a Poisson random variable, what is the probability that no flaws occur in a certain portion of wire of length 5 millimeters? What is the mean number of flaws in
5.70 A company purchases large lots of a certain kind of electronic device. A method is used that rejects a lot if 2 or more defective units are found in a random sample of 100 units.(a) What is the mean number of defective units found in a sample of 100 units if the lot is 2% defective?(b) What is
5.69 The probability that a person will die when he or she contracts a viral infection is 0.004. Of the next 4000 people infected, what is the mean number who will die?
5.68 Consider Exercise 5.62. What is the mean number of students who fail the test?
5.67 The number of customers arriving per hour at a certain automobile service facility is assumed to follow a Poisson distribution with mean λ = 7.(a) Compute the probability that more than 10 customers will arrive in a 2-hour period.(b) What is the mean number of arrivals during a 2-hour period?
5.66 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, according to a Poisson process, at the rate of 6 per hour. Thus, the Poisson parameter for arrivals
5.65 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 number of cars per year that will experience the catastrophe is a Poisson random variable with λ =
5.64 Find the mean and variance of the random variable X in Exercise 5.61, representing the number of persons among 10,000 who make an error in preparing their income tax returns.
5.63 Find the mean and variance of the random variable X in Exercise 5.58, representing the number of hurricanes per year to hit a certain area of the eastern United States.
5.62 The probability that a student at a local high school fails the screening test for scoliosis (curvature of the spine) is known to be 0.004. Of the next 1875 students at the school who are screened for scoliosis, find the probability that(a) fewer than 5 fail the test;(b) 8, 9, or 10 fail the
5.61 Suppose that, on average, 1 person in 1000 makes a numerical error in preparing his or her income tax return. If 10,000 returns are selected at random and examined, find the probability that 6, 7, or 8 of them contain an error.
5.60 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 inspected.
5.59 Suppose the probability that any given person will believe a tale about the transgressions of a famous actress is 0.8. What is the probability that(a) the sixth person to hear this tale is the fourth one to believe it?(b) the third person to hear this tale is the first one to believe it?
5.58 A certain area of the eastern United States is, on average, hit by 6 hurricanes a year. Find the probability that in a given year that area will be hit by(a) fewer than 4 hurricanes;(b) anywhere from 6 to 8 hurricanes.
5.57 On average, a textbook author makes two wordprocessing errors per page on the first draft of her textbook.What is the probability that on the next page she will make(a) 4 or more errors?(b) no errors?
5.56 On average, 3 traffic accidents per month occur at a certain intersection. What is the probability that in any given month at this intersection(a) exactly 5 accidents will occur?(b) fewer than 3 accidents will occur?(c) at least 2 accidents will occur?
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