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descriptive statistics
Probability And Statistics For Engineers And Scientists 9th Edition Ronald E. Walpole; Raymond H. Myers; Sharon L. Myers; Keying Ye - Solutions
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 probability that between 482 and 510, inclusive, of the next 1000 randomly selected Valium users
Show that the failure-rate function is given by Z(t) = αβtβ−1, t> 0, if and only if the time to failure distribution is the Weibull distribution f(t) = αβtβ−1 e−αtβ, t> 0.
Consider the information in Exercise 6.58.(a) What is the probability that more than 1 minute elapses between arrivals?(b) What is the mean number of minutes that elapse between arrivals?
The number of automobiles that arrive at a certain intersection per minute has a Poisson distribution with a mean of 5. Interest centers around the time that elapses before 10 automobiles appear at the intersection.(a) What is the probability that more than 10 automobiles appear at the intersection
For Exercise 6.56, what is the mean power usage(average dB per hour)? What is the variance?
Rate data often follow a lognormal distribution.Average power usage (dB per hour) for a particular company is studied and is known to have a lognormal distribution with parameters μ = 4 and σ = 2. What is the probability that the company uses more than 270 dB during any particular hour?
Computer response time is an important application of the gamma and exponential distributions.Suppose that a study of a certain computer system reveals that the response time, in seconds, has an exponential distribution with a mean of 3 seconds.(a) What is the probability that response time exceeds
The lifetime, in weeks, of a certain type of transistor is known to follow a gamma distribution with mean 10 weeks and standard deviation √50 weeks.(a) What is the probability that a transistor of this type will last at most 50 weeks?(b) What is the probability that a transistor of this type will
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
Derive the mean and variance of the Weibull distribution.
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.
If the proportion of a brand of television set 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
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.
Derive the mean and variance of the beta distribution.
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?
The life, in years, of a certain type of electrical switch has an exponential distribution with an average life β = 2. If 100 of these switches are installed in different systems, what is the probability that at most 30 fail during the first year?
The length of time for one individual to be served at a cafeteria is a random variable having an exponential distribution with a mean of 4 minutes. What is the probability that a person is served in less than 3 minutes on at least 4 of the next 6 days?
In a certain city, the daily consumption of electric power, in millions of kilowatt-hours, is a random variable X having a gamma distribution with meanμ = 6 and variance σ2 = 12.(a) Find the values of α and β.(b) Find the probability that on any given day the daily power consumption will exceed
(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?
Suppose that the time, in hours, required to repair a heat pump is a random variable X having a gamma distribution with parameters α = 2 andβ = 1/2. What is the probability that on the next service call(a) at most 1 hour will be required to repair the heat pump?(b) at least 2 hours will be
If a random variable X has the gamma distribution with α = 2 and β = 1, find P(1.8
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?
Use the gamma function with y = √2x to show that Γ(1/2) =√π.
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 the
The serum cholesterol level X in 14-year-old boys has approximately a normal distribution with mean 170 and standard deviation 30.(a) Find the probability that the serum cholesterol level of a randomly chosen 14-year-old boy exceeds 230.(b) In a middle school there are 300 14-year-old boys.Find the
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 seats, a
A company produces component parts for an engine. Parts specifications suggest that 95% of items meet specifications. The parts are shipped to customers in lots of 100.(a) What is the probability that more than 2 items in a given lot will be defective?(b) What is the probability that more than 10
A pair of dice is rolled 180 times. What is the probability that a total of 7 occurs(a) at least 25 times?(b) between 33 and 41 times inclusive?(c) exactly 30 times?
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 drunk
A pharmaceutical company knows that approximately 5% of its birth-control pills have an ingredient that is below the minimum strength, thus rendering the pill ineffective. What is the probability that fewer than 10 in a sample of 200 pills will be ineffective?
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?
A drug manufacturer claims that a certain drug cures a blood disease, on the average, 80% of the time.To check the claim, government testers use the drug on a sample of 100 individuals and decide to accept the claim if 75 or more are cured.(a) What is the probability that the claim will be rejected
If 20% of the residents in a U.S. city prefer a white telephone over any other color available, what is the probability that among the next 1000 telephones installed in that city(a) between 170 and 185 inclusive will be white?(b) at least 210 but not more than 225 will be white?
Researchers at George Washington University and the National Institutes of Health claim that approximately 75% of people believe “tranquilizers work very well to make a person more calm and relaxed.” Of the next 80 people interviewed, what is the probability that(a) at least 50 are of this
The probability that a patient recovers from a delicate heart operation is 0.9. Of the next 100 patients having this operation, what is the probability that(a) between 84 and 95 inclusive survive?(b) fewer than 86 survive?
A process yields 10% defective items. If 100 items are randomly selected from the process, what is the probability that the number of defectives(a) exceeds 13?(b) is less than 8?
A process for manufacturing an electronic component yields items of which 1% are defective. A quality control plan is to select 100 items from the process, and if none are defective, the process continues. Use the normal approximation to the binomial to find(a) the probability that the process
A coin is tossed 400 times. Use the normal curve approximation to find the probability of obtaining(a) between 185 and 210 heads inclusive;(b) exactly 205 heads;(c) fewer than 176 or more than 227 heads.
The IQs of 600 applicants to a certain college are approximately normally distributed with a mean of 115 and a standard deviation of 12. If the college requires an IQ of at least 95, how many of these students will be rejected on this basis of IQ, regardless of their other qualifications? Note that
If a set of observations is normally distributed, what percent of these differ from the mean by(a) more than 1.3σ?(b) less than 0.52σ?
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 these
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 kilograms;(b)
A company pays its employees an average wage of $15.90 an hour with a standard deviation of $1.50. If the wages are approximately normally distributed and paid to the nearest cent,(a) what percentage of the workers receive wages between $13.75 and $16.22 an hour inclusive?(b) the highest 5% of the
The heights of 1000 students are normally distributed with a mean of 174.5 centimeters and a standard deviation of 6.9 centimeters. Assuming that the heights are recorded to the nearest half-centimeter, how many of these students would you expect to have heights(a) less than 160.0 centimeters?(b)
The average life of a certain type of small motor is 10 years with a standard deviation of 2 years. The manufacturer replaces free all motors that fail while under guarantee. If she is willing to replace only 3% of the motors that fail, how long a guarantee should be offered? Assume that the
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 the
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 hour?(b)
The finished inside diameter of a piston ring is normally distributed with a mean of 10 centimeters and a standard deviation of 0.03 centimeter.(a) What proportion of rings will have inside diameters exceeding 10.075 centimeters?(b) What is the probability that a piston ring will have an inside
A research scientist reports that mice will live an average of 40 months when their diets are sharply restricted and then enriched with vitamins and proteins.Assuming that the lifetimes of such mice are normally distributed with a standard deviation of 6.3 months, find the probability that a given
The loaves of rye bread distributed to local stores by a certain bakery have an average length of 30 centimeters and a standard deviation of 2 centimeters.Assuming that the lengths are normally distributed, what percentage of the loaves are(a) longer than 31.7 centimeters?(b) between 29.3 and 33.5
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 a cup
According to Chebyshev’s theorem, the probability that any random variable assumes a value within 3 standard deviations of the mean is at least 8/9. 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 P(μ −
Given the normally distributed variable X with mean 18 and standard deviation 2.5, find(a) P(X < 15);(b) the value of k such that P(Xk)=0.1814;(d) P(17
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 x
Given a standard normal distribution, find the value of k such that(a) P(Z>k)=0.2946;(b) P(Z
Find the value of z if the area under a standard normal curve(a) to the right of z is 0.3622;(b) to the left of z is 0.1131;(c) between 0 and z, with z > 0, is 0.4838;(d) between −z and z, with z > 0, is 0.9500.
Given a standard normal distribution, find the area under the curve that lies(a) to the left of z = −1.39;(b) to the right of z = 1.96;(c) between z = −2.16 and z = −0.65;(d) to the left of z = 1.43;(e) to the right of z = −0.89;(f) between z = −0.48 and z = 1.74.
A bus arrives every 10 minutes at a bus stop. 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 7 minutes?(b) What is the probability that the individual waits
The daily amount of coffee, in liters, dispensed by a machine located in an airport lobby is a random variable X having a continuous uniform distribution with A = 7 and B = 10. Find the probability that on a given day the amount of coffee dispensed by this machine will be(a) at most 8.8 liters;(b)
Suppose X follows a continuous uniform distribution from 1 to 5. Determine the conditional probability P(X > 2.5 | X ≤ 4).
Given a continuous uniform distribution, show that(a) μ = A+B 2 and(b) σ2 = (B−A)2 12 .
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 each
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., 0.0001).
There are two vacancies in a certain university statistics department. Five 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 two chosen
Go back to Review Exercise 5.95(a). Recompute the probability using the binomial distribution. Comment.
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
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 incoming
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 restrictions,
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 rare
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% defective?
It is known by researchers that 1 in 100 people carries 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 7 individuals carry the gene? Use a Poisson approximation.Again, using the approximation, what
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 fourth child?
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?
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
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 the
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 0.990. Let us assume that the outcomes of attempted starts are independent.(a) What is the probability that the engine
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 0.03 and the board contains 200 diodes.(a) What is the mean number of failures among the diodes?(b) What is the variance?(c)
Suppose that out of 500 lottery tickets sold, 200 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.
(a) Suppose that you throw 4 dice. Find the probability that you get at least one 1.(b) Suppose that you throw 2 dice 24 times. Find the probability that you get at least one (1, 1), that is,“snake-eyes.”
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.
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 1%defective?(b) What is the probability of accepting a
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
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 occurrence,
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.
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 Honda,
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 be
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
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 before a
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?
It is known that 3% 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 people
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
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 applies to
For a certain type of copper wire, it is known that, on the average, 1.5 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 a
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 1% defective?(b) What is the
The probability that a person will die when he or she contracts a virus infection is 0.001. Of the next 4000 people infected, what is the mean number who will die?
Consider Exercise 5.62. What is the mean number of students who fail the test?
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?
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 over
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.(a)
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.
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