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probability statistics
Probability And Statistics For Engineers And Scientists 9th Edition Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, Keying E. Ye - Solutions
The probability of success of 900 students appearing in a test independently is 0.50. Using normal approximation, find the probability that 1. between 415 and 500 students pass, both inclusive;2. exactly 470 students pass;3. fewer than 480 and more than 420 students pass.
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 to
If a set of observations is normally distributed, what percent of these differ from the mean by, 1. More than 1.50σ;2. Less than 0.75σ.
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.1. 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 1. over 9.5 kilograms;2. of
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, 1. what percentage of the operating hours will have an expense of less than $16.50 per hour?2. what will be the
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 1. less than 55 kilograms?2. between 60
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% of
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.1. 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.1. What is the probability that a trip will take at least 1/2 hour?2.
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.1. What percentage of the washers have an inside diameter of more than 0.61 centimeters?2. Obtain the probability that the inside
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 a
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 1. longer than 5.05 centimeters?2. between 4.95 and 5.05 centimeters in length?3. shorter than 4.90 centimeters?
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, 1. what fraction of the cups will contain more than 224 milliliters?2. what is the probability that a cup
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 σ , what is the exact value of P(μ
Given the normally distributed variable X with a mean of 20 and standard deviation of 2, find 1. P(X < 16);2. the value of k such that P(X < k) = 0.4090;3. the value of k such that P(X > k) = 0.8599;4. P(17 < X < 22).
Given a normal distribution with μ = 30 and σ = 6, find 1. the normal curve area to the right of x = 17;2. the normal curve area to the left of x = 22;3. the normal curve area between x = 32 and x = 41;4. the value of x that has 80% of the normal curve area to the left;5. the two values of x that
Given the standard normal distribution, find the value of k such that 1. P(Z > k) = 0.9625;2. P(Z < k) = 0.6255;3. P(0.17 < Z < k) = 0.367.
Find the value of z if the area under a standard normal curve 1. to the right of z is 0.3745;2. to the left of z is 0.3050;3. between –z and 0, with –z < 0, is 0.4838;4. between –z and z with z > 0, is 0.90.
Given a standard normal distribution, find the area under the curve that lies 1. to the left of z = –1.10;2. to the right of z = 1.645;3. between z = –2.43 and –0.45;4. to the left of z = 0.45;5. to the right of z = –0.45;6. between z = –0.45 and z = 0.45.
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.1. What is the probability that the individual waits more than 10 minutes?2. What is the probability that the individual waits
The daily milk consumption, in thousands of liters, dispensed by a milk booth located in a city is a random variable, X, having a continuous uniform distribution with A = 7 66 and B = 10. Find the probability that on a given day, the amount of milk dispensed by this booth will be 1. at most 8.8
Suppose X follows a continuous uniform distribution from 0 to 5. Determine the conditional probability, P(X > 2.6 | X ≥ 1).
Given a continuous uniform distribution, show that 1. and 2.
Group Project: Divide the class into two groups of approximately equal size. The students in group 1 will each toss a coin 10 times (n ) and count the number of heads obtained.The students in group 2 will each toss a coin 40 times (n ) and again count the number of heads. The students in each group
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. Six individuals apply. Three have expertise in linear models, and one has expertise in applied probability. The search committee is instructed to choose the two applicants randomly.1. 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.1. 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?2. Suppose it is decided
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
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
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.1. What is your response to the presumption that the process is 1% defective?
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 21 individuals carry the gene? Use a Poisson approximation. Again, using the approximation, what is
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 0.25.1. What is the probability that the driller drills at 10 locations and has 1 success?2. The driller
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 0.001.1. What is the probability that a lot is accepted?2. 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 0.99. Let us assume that the outcomes of attempted starts are independent.1. What is the probability that the engine is
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 100 diodes.1. What is the mean number of failures among the diodes?2. What is the variance?3. The
Suppose that out of 500 lottery tickets sold, 100 pay off at least the cost of the ticket. Now suppose that you buy 10 tickets.Find the probability that you will win back at least the cost of 3 tickets.
1. Suppose that you throw 4 dice. Find the probability that you get at least one 2.2. Suppose that you throw 2 dice 10 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.1. Find the probability that in a given 3-minute period nobody enters the store.2. 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.1. What is the probability of rejecting a lot that is 3% defective?2. 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.1. 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 1. no more than 4 calls come in any minute;2. fewer than 2 calls come in any minute;3. more than 10 calls come in a 5-minute period.
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.1. Comment on the validity of the report. Use a probability in your argument.2. 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 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 10 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.1. What is the mean number of defective units found in a sample of 100 units if the lot is 3% defective?2. What is the
The probability that a person will die when he or she contracts a viras infection is 0.003. Of the next 3000 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 = 7.1. Compute the probability that more than 10 customers will arrive in a 2-hour period.2. 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 = 5.1.
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.5.64 Find the mean and variance of the random variable X in Exercise 5.61, representing the number of persons among 10,000 who make
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 1. fewer than 5 fail the test;2. 8, 9, or 10 fail the test.
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.
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 1. on a given acre;2. on 2 of the next 3 acres inspected.
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 1. the sixth person to hear this tale is the fourth one to believe it?2. the third person to hear this tale is the first one to believe it?
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 1. fewer than 4 hurricanes;2. anywhere from 6 to 8 hurricanes.
On average, a textbook author makes two word-processing errors per page on the first draft of her textbook. What is the probability that on the next page she will make 1. 4 or more errors?2. no errors?
On average, 3 traffic accidents per month occur at a certain intersection. What is the probability that in any given month at this intersection 1. exactly 5 accidents will occur?2. fewer than 3 accidents will occur?3. at least 2 accidents will occur?
The probability that a student pilot passes the written test for a private pilot’s license is 0.7. Find the probability that a given student will pass the test 1. on the third try;2. before the fourth try.
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 a valid estimate, find the probability that on a given day the fifth prescription written by a
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 1. more than 5 times?2. not at all?
A scientist inoculates 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 1/5, what is the probability that 6 mice are required?
Three people toss a fair coin and the odd one pays for coffee. If the coins all turn up the same, they are tossed again.Find the probability that fewer than 2 tosses are needed.
Find the probability that a person flipping a coin gets 1. the third head on the seventh flip;2. the first head on the fourth flip.
The probability that a person living in a certain city owns a dog is estimated to be 0.4. Find the probability that the seventh person randomly interviewed in that city is the fifth one to own a dog.
Every hour, 10,000 cans of soda are filled by a machine, among which 100 underfilled cans are produced. Each hour, a sample of 30 cans is randomly selected and the number of ounces of soda per can is checked. Denote by X the number of cans selected that are underfilled. Find the probability that at
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 but not all can be inspected.Suppose that 3 of the firms are in violation.1. What is the probability
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 selected and all are tested. Suppose there are 2 faulty compressors in the batch of 15.1. What is the
Biologists doing studies in a particular environment often tag and release subjects in order to estimate the size of a population or the prevalence of certain features in the population. Ten animals of a certain population thought to be extinct (or near extinction) are caught, tagged, and released
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 red ball are selected.
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 1. all nationalities are represented;2. all nationalities except Italian are represented.
A nationwide survey of 17,000 college 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 opinion, what is the probability that more than 8 but fewer than 12 disapprove of smoking pot daily?
An annexation suit against a county subdivision of 1200 residences is being considered by a neighboring city. If the occupants of half the residences object to being annexed, what is the probability that in a random sample of 10 at least 3 favor the annexation suit?
Among 150 IRS employees in a large city, only 30 are women. If 20 of the employees are chosen at random to provide free tax assistance for the residents of this city, use the binomial approximation to the hypergeometric distribution to find the probability that at least 2 women are selected.
Suppose that the manufacturing company of Exercise 5.36 decides to change its acceptance scheme. Under the new scheme, an inspector takes 1 item at random, inspects it, and then replaces it in the box; a second inspector does likewise.Finally, a third inspector goes through the same procedure. The
A manufacturing company uses an acceptance scheme on items from a production line before they are shipped. The plan is a two-stage one. Boxes of 25 items are readied for shipment, and a sample of 3 items is tested for defectives. If any defectives are found, the entire box is sent back for 100%
A company is interested in evaluating its current inspection procedure for 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 be defective. What proportion of shipments with 20% defectives will be accepted?
If 7 cards are dealt from an ordinary deck of 52 playing cards, what is the probability that 1. exactly 2 of them will be face cards?2. at least 1 of them will be a queen?
From a lot of 9 missiles, 3 are selected at random and fired.If the lot contains 4 defective missiles that will not fire, what is the probability that 1. all 3 will fire?2. at most 2 will not fire?
A random committee of size 4 is selected from 5 doctors and 2 nurses. Write a formula for the probability distribution of the random variable X representing the number of doctors on the committee. Find P(2 ≤ X ≤ 3).
To avoid detection at customs, a traveler places 5 narcotic tablets in a bottle containing 10 vitamin tablets that are similar in appearance. If the customs official selects 3 of the tablets at random for analysis, what is the probability that the traveler will be arrested for illegal possession of
A homeowner plants 5 bulbs selected at random from a box containing 5 tulip bulbs and 4 daffodil bulbs. What is the probability that he planted 3 daffodil bulbs and 2 tulip bulbs?
A manufacturer knows that on average 20% of the electric toasters produced require repairs within 1 year after they are sold. When 20 toasters are randomly selected, find appropriate numbers x and y such that 1. the probability that at least x of them will require repairs is less than 0.5;2. 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 1. exactly 18 will have a useful life of at least 800 hours;2. at least 15 will have a useful life of at least 800 hours;3. at least 2 will not have a useful
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 1. by using the formula for the binomial distribution;2. by using Table A.1.
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, find the probability that at most 3 chips fail in a random sample of 20.
A safety engineer claims that only 40% of all workers wear safety helmets when they eat lunch at the workplace. Assuming that this claim is right, find the probability that 4 of 6 workers randomly chosen will be wearing their helmets while having lunch at the workplace.
The probabilities are 0.4, 0.3, 0.1, and 0.2, respectively, that a delegate to a certain convention arrived by air, bus, automobile, or train. What is the probability that among 10 delegates randomly selected at this convention, 3 arrived by air, 3 arrived by bus, 2 arrived by automobile, and 2
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 parts such that a person throwing a dart that lands in a specific region scores the value of the
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