Question: Consider the information in Review Exercise 5.

> 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 ineffe

> 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 fro

> 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

> Let X1 and X2 be discrete random variables with the joint multinomial distribution for x1 = 0, 1, 2; x2 = 0, 1, 2; x1 + x2 ≤ 2; and zero elsewhere. Find the joint probability distribution of Y1 = X1 + X2 and Y2 = X1 âˆ&#1

> 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 m

> 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 probabi

> A random sample of 12 shearing pins is taken in a study of the Rockwell hardness of the pin head. Measurements on the Rockwell hardness are made for each of the 12, yielding an average value of 48.50 with a sample standard deviation of 1.5. Assuming the

> 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

> 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.

> If both X and Y, distributed independently, follow exponential distributions with mean parameter 1, find the distributions of (a) U = X + Y; (b) V = X/(X + Y ).

> Using the moment-generating function of Exercise 7.21, show that the mean and variance of the chisquared distribution with v degrees of freedom are, respectively, v and 2v. Exercise 7.21: Show that the moment-generating function of the random variable X

> Show that the moment-generating function of the random variable X having a chi-squared distribution with v degrees of freedom is MX(t) = (1 − 2t) −v/2.

> The moment-generating function of a certain Poisson random variable X is given by MX(t) = e4(et−1). Find P(μ − 2σ

> Let X be a binomial random variable with probability distribution Find the probability distribution of the random variable Y = X2. 3-r x = 0, 1, 2, 3, f(r) = 0, elsewhere.

> A random variable X has the Poisson distribution p(x; μ) = e −μμx/x! for x = 0, 1, 2, . . . . Show that the moment-generating function of X is MX(t) = eμ(et−1). Using MX(t), find the mean and variance of the Poisson distribution.

> A random sample of 10 chocolate energy bars of a certain brand has, on average, 230 calories per bar, with a standard deviation of 15 calories. Construct a 99% confidence interval for the true mean calorie content of this brand of energy bar. Assume that

> A random variable X has the geometric distribution g(x; p) = pqx−1 for x = 1, 2, 3, . . . . Show that the moment-generating function of X is and then use MX(t) to find the mean and variance of the geometric distribution. pet 1- qe

> A random variable X has the discrete uniform distribution Show that the moment-generating function of X is SE r = 1,2, ..., k, 0, elsewhere. f (x; k) = e'(1 – ekt) k(1 – et) Mx (t) =

> Show that the rth moment about the origin of the gamma distribution is B"T(a + r) T(a)

> Let X have the probability distribution Find the probability distribution of the random variable Y = X2. 2(x+1). -1< x < 2, f(x) = elsewhere.

> Let X be a random variable with probability distribution Find the probability distribution of the random variable Y = X2. S, -1< x < 1, 10, 1+z f(r) = elsewhere.

> A current of I amperes flowing through a resistance of R ohms varies according to the probability distribution If the resistance varies independently of the current according to the probability distribution find the probability distribution for the p

> A certain supplier manufactures a type of rubber mat that is sold to automotive companies. The material used to produce the mats must have certain hardness characteristics. Defective mats are occasionally discovered and rejected. The supplier claims that

> A group of human factor researchers are concerned about reaction to a stimulus by airplane pilots in a certain cockpit arrangement. An experiment was conducted in a simulation laboratory, and 15 pilots were used with average reaction time of 3.2 seconds

> Consider the statistic S2 p, the pooled estimate of σ2 discussed in Section 9.8. It is used when one is willing to assume that σ2 1 = σ22 = σ2. Show that the estimator is unbiased for σ2 [i.e., show that E(S2p) = σ2]. You may make use of results from any

> A consumer group is interested in comparing operating costs for two different types of automobile engines. The group is able to find 15 owners whose cars have engine type A and 15 whose cars have engine type B. All 30 owners bought their cars at roughly

> A machine produces metal pieces that are cylindrical in shape. A sample of pieces is taken, and the diameters are found to be 1.01, 0.97, 1.03, 1.04, 0.99, 0.98, 0.99, 1.01, and 1.03 centimeters. Find a 99% confidence interval for the mean diameter of pi

> A machine is used to fill boxes with product in an assembly line operation. Much concern centers around the variability in the number of ounces of product in a box. The standard deviation in weight of product is known to be 0.3 ounce. An improvement is i

> A manufacturer turns out a product item that is labeled either “defective” or “not defective.” In order to estimate the proportion defective, a random sample of 100 items is taken from production, and 10 are found to be defective. Following implementatio

> Refer to Review Exercise 9.106. Suppose there is concern about whether the point estimate ˆp = 8/30 is accurate enough because the confidence interval around p is not sufficiently narrow. Using ˆp as the estimate of p, how many companies would need to be

> A random sample of 30 firms dealing in wireless products was selected to determine the proportion of such firms that have implemented new software to improve productivity. It turned out that 8 of the 30 had implemented such software. Find a 95% confidenc

> A labor union is becoming defensive about gross absenteeism by its members. The union leaders had always claimed that, in a typical month, 95% of its members were absent less than 10 hours. The union decided to check this by monitoring a random sample of

> Consider Review Exercise 9.103. Let us assume that the data have not been collected yet and that previous statistics suggest that σ1 = σ2 = $4000. Are the sample sizes in Review Exercise 9.103 sufficient to produce a 95% confidence interval on μ1 −μ2 hav

> A survey was done with the hope of comparing salaries of chemical plant managers employed in two areas of the country, the northern and west central regions. An independent random sample of 300 plant managers was selected from each of the two regions. Th

> Specify the moment estimators for α and β for the gamma distribution.

> Specify the moment estimators for μ and σ2 for the lognormal distribution.

> Specify the moment estimators for μ and σ2 for the normal distribution.

> A random sample of 12 graduates of a certain secretarial school typed an average of 79.3 words per minute with a standard deviation of 7.8 words per minute. Assuming a normal distribution for the number of words typed per minute, find a 95% confidence in

> A UCLA researcher claims that the life span of mice can be extended by as much as 25% when the calories in their diet are reduced by approximately 40% from the time they are weaned. The restricted diet is enriched to normal levels by vitamins and protein

> 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 throug

> 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 probabili

> 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 f

> 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 comp

> 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 re observed. (a) What is your response to the p

> 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 approxi

> 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?

> 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

> In testing a certain kind of truck tire over 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 that (a) from 3 to 6 have blowouts; (b) fewer than 4 have

> 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 probabil

> 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.

> 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 am

> 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

> 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% defecti

> 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 dur

> 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) W

> 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

> 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 problems. Find the probability that among the next 8 users from

> 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

> 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

> 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 f

> 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 e

> 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 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 ob

> 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

> 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 i

> 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 mill

> 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 uni

> 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, fewer than half are chain smokers; (b) find the probabili

> 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? Exercise 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 st

> 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 n

> 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. T

> 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 catastroph

> 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.

> 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.

> 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

> 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 (a) on a given acre; (b) on 2 of the next 3 acres inspected.

> 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 that among 6 companies surveyed at random, the number t

> 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

> 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.

> 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 (a) 4 or more errors? (b) 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 (a) exactly 5 accidents will occur? (b) fewer than 3 accidents will occur? (c) 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 (a) on the third try; (b) 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

> 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 5 times? (b) 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 1/6, what is the probability that 8 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 4 tosses are needed.

> 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.

> According to Chemical Engineering Progress (November 1990), approximately 30% of all pipework failures in chemical plants are caused by operator error. (a) What is the probability that out of the next 20 pipework failures at least 10 are due to operator

> 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.

> Every hour, 10,000 cans of soda are filled by a machine, among which 300 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

> 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 fi

> 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 com

> Biologists doing studies in a particular environment often tag and release subjects in order to estimate 158 Chapter 5 Some Discrete Probability Distributions the size of a population or the prevalence of certain features in the population. Ten animals o