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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
3.37 Determine the values of c so that the following functions represent joint probability distributions of the random variables, X and Y :(a) f(x, y) = cxy, for x = 1, 2; y = 1, 2;(b) f(x, y) = c|x − y|, for x = −2, 0, 2; y = −3, 3.
3.36 For a laboratory assignment, if the equipment is working, the density function of the observed outcome, X, is f(x) =2(1 − x), 0 < x < 1, 0, otherwise.(a) Calculate P(X < 0.5).(b) What is the probability that X will exceed 0.4?(c) Given that X ≥ 0.5, what is the probability that X will be
3.35 Suppose it is known from large amounts of historical data that X, the number of cars that arrive at a specific intersection during a 20-second time period, is characterized by the following discrete probability function:f(x) = e−4 4x x! , for x = 0, 1, 2, . . . .(a) Find the probability that
3.34 Magnetron tubes are produced on an automated assembly line. A sampling plan is used periodically to assess quality of the lengths of the tubes. This measurement is subject to uncertainty. It is thought that the probability that a random tube meets length specification is 0.99. A sampling plan
3.33 Suppose a certain type of small data processing firm is so specialized that some have difficulty making a profit in their first year of operation. The probability density function that characterizes the proportion Y that makes a profit is given by f(y) =ky2 (1 − y)3 , 0 ≤ y ≤ 1, 0,
3.32 The proportion of the budget for a certain type of industrial company that is allotted to environmental and pollution control is coming under scrutiny. A data collection project determines that the distribution of these proportions is given by f(y) =5(1 − y)4 , 0 ≤ y ≤ 1, 0,
3.31 Based on extensive testing, it is determined by the manufacturer of a washing machine that the time Y (in years) before a major repair is required is characterized by the probability density function f(y) =14 e−y /4, y≥ 0, 0, elsewhere.(a) Critics would certainly consider the product a
3.30 Measurements of scientific systems are always subject to variation, some more than others. There are many structures for measurement error, and statisticians spend a great deal of time modeling these errors.Suppose the measurement error X of a certain physical quantity is decided by the
3.29 An important factor in solid missile fuel is the particle size distribution. Significant problems occur if the particle sizes are too large. From production data in the past, it has been determined that the particle size (in micrometers) distribution is characterized by f(x) =3x−4, x>1, 0,
3.28 A cereal manufacturer is aware that the weight of the product in the box varies slightly from box to box. In fact, considerable historical data have allowed the determination of the density function that describes the probability structure for the weight (in ounces). Letting X be the random
3.27 The time to failure in hours of an important piece of electronic equipment used in a DVD player has the density function f(x) = 1 1000 exp(−x/1000), x≥ 0, 0, x< 0.(a) Find F(x).(b) Determine the probability that the component (and thus the DVD player) lasts more than 1000 hours before the
3.26 From a box containing 3 black balls and 2 green balls, 3 balls are drawn in succession, each ball being replaced in the box before the next draw is made. Find the probability distribution for the number of green balls.
3.25 From a box containing 4 dimes and 2 nickels, 3 coins are selected at random without replacement.Find the probability distribution for the total T of the 3 coins. Express the probability distribution graphically as a probability histogram.
3.24 Find the probability distribution for the number of comic books when 4 books are selected at random from a collection consisting of 5 comic books, 2 art books, and 3 math books. Express your results by means of a formula.
3.23 Find the cumulative distribution function of the random variable W in Exercise 3.8. Using F(w), find(a) P(W >0);(b) P(−1 ≤W
3.22 Two cards are drawn in succession from a deck without replacement. Find the probability distribution for the number of spades.
3.21 Consider the density function f(x) =k√x, 0 ≤ x < 4, 0, elsewhere.(a) Evaluate k.(b) Find F(x) and use it to evaluate P(3 < X < 4).
3.20 For the density function of Exercise 3.18, find F(x), and use it to evaluate P(3 ≤X < 4).
3.19 For the density function of Exercise 3.17, find F(x). Use it to evaluate P(2 < X < 2.5).
3.18 A continuous random variable X that can assume values between x = 2 and x = 5 has a density function given by f(x) = 2(1+x)/27. Find(a) P(X < 4);(b) P(3 ≤X < 4).
3.17 A continuous random variable X that can assume values between x = 1 and x = 3 has a density function given by f(x) = 1/2.(a) Show that the area under the curve is equal to 1.(b) Find P(2 < X < 2.5).(c) Find P(X ≤ 1.6).
3.16 Construct a graph of the cumulative distribution function of Exercise 3.15.
3.15 Find the cumulative distribution function of the random variable X representing the number of defectives in Exercise 3.11. Then using F(x), find(a) P(X = 1);(b) P(0 < X ≤ 2).
3.14 The waiting time, in hours, between successive speeders spotted by a radar unit is a continuous random variable with a cumulative distribution function F(x) =0, x< 0, 1 − e−4x, x≥ 0.Find the probability of waiting fewer than 12 minutes between successive speeders(a) using the cumulative
3.13 The probability distribution of X, the number of imperfections per 10 meters of a synthetic fabric in continuous rolls of uniform width, is given by x 0 1 2 3 4 f(x) 0.41 0.37 0.16 0.05 0.01 Construct the cumulative distribution function of X.
3.12 An investment firm offers its customers municipal bonds that mature after varying numbers of years.Given that the cumulative distribution function of T , the number of years to maturity for a randomly selected bond, is F(t) =⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩0, t 3);(c) P(1.4 < T < 6);(d)
3.11 A shipment of 7 televisions contains 2 defective ones. A hotel makes a random purchase of 3 televisions. If x is the number of defective ones purchased by the hotel, find the probability distribution of X. Express the results graphically as a probability histogram.
3.10 Find a formula for the probability distribution of the random variable X representing the outcome when a number is drawn at random from a hat having numbers from 0 to 9.
3.9 The proportion of people who respond to a certain mail-order solicitation is a continuous random variable X that has the density function f(x) =2(x+2)5 , 0 < x < 1, 0, elsewhere.(a) Show that P(0 < X < 1) = 1.(b) Find the probability that more than 1/4 but fewer than 3/4 of the people
3.8 Find the probability distribution of the random variable W in Exercise 3.3, assuming that the coin is biased so that a head is twice as likely to occur as a tail.
3.7 The total number of hours, measured in units of 100 hours, that a family runs a vacuum cleaner over a period of one year is a continuous random variable X that has the density function f(x) =⎧⎨⎩x, 0 < x < 1, 2 − x, 1 ≤ x < 2, 0, elsewhere.Find the probability that over a period of one
3.6 The shelf life, in days, for bottles of a certain prescribed medicine is a random variable having the density function f(x) =20,000(x+100)3, x>0, 0, elsewhere.Find the probability that a bottle of this medicine will have a shell life of(a) at least 200 days;(b) anywhere from 80 to 120 days.
3.5 Determine the value c so that each of the following functions can serve as a probability distribution of the discrete random variable X:(a) f(x) = c(x2 + 4), for x = 0, 1, 2;(b) f(x) = c3 x 3 3−x, for x = 0, 1, 2.
3.4 A coin is flipped until 2 heads occur in succession.List only those elements of the sample space that require 6 or fewer tosses. Is this a discrete sample space?Explain.
3.3 Let W be a random variable giving the number of heads minus the number of tails in three tosses of a coin. List the elements of the sample space S for the three tosses of the coin and to each sample point assign a value w of W.
3.2 An overseas shipment of 5 foreign automobiles contains 2 that have slight paint blemishes. If an agency receives 2 of these automobiles at random, list the elements of the sample space, S, using the letters B and N for blemished and unblemished units, respectively. Then, to each sample point,
3.1 Classify the following random variables as discrete or continuous:X: the number of automobile accidents per year in Virginia.Y : the length of time to play 18 holes of golf.M: the amount of milk produced yearly by a particular cow.N: the number of eggs laid each month by a hen.P: the number of
2.128 Group Project: Give each student a bag of chocolate M&Ms. Divide the students into groups of 5 or 6. Calculate the relative frequency distribution for color of M&Ms for each group.(a) What is your estimated probability of randomly picking a yellow? a red?(b) Redo the calculations for the
2.127 There is a 50-50 chance that the queen carries the gene of hemophilia. If she is a carrier, then each prince has a 50-50 chance of having hemophilia independently.If the queen is not a carrier, the prince will not have the disease. Suppose the queen has had three princes without the disease.
2.126 During bad economic times, industrial workers are dismissed and are often replaced by machines. The history of 100 workers whose loss of employment is attributable to technological advances is reviewed. For each of these individuals, it is determined if he or she was given an alternative job
2.125 A survey of those using a particular statistical software system indicated that 10% were dissatisfied.Half of those dissatisfied purchased the system from vendor A. It is also known that 20% of those surveyed purchased from vendor A. Given that the software was purchased from vendor A, what
2.124 A firm is accustomed to training operators who do certain tasks on a production line. Those operators who attend the training course are known to be able to meet their production quotas 90% of the time. New operators who do not take the training course only meet their quotas 65% of the time.
2.123 An industrial plant is conducting a study to determine how quickly injured workers are back on the job following injury. Records show that 10% of all injured workers are admitted to the hospital for treatment and 15% are back on the job the next day. In addition, studies show that 2% are both
2.122 In the field of quality control, the science of statistics is often used to determine if a process is out of control. Suppose the process is, indeed, out of control and 10% of items produced are defective.(a) If three items arrive off the process line in succession, what is the probability
2.121 A construction company employs two sales engineers.Engineer 1 does the work of estimating cost for 70% of jobs bid by the company. Engineer 2 does the work for 30% of jobs bid by the company. It is known that the error rate for engineer 1 is such that 0.02 is the probability of an error when
2.120 A rare disease exists with which only 1 in 500 is affected. A test for the disease exists, but of course it is not infallible. A correct positive result (patient actually has the disease) occurs 95% of the time, while a false positive result (patient does not have the disease) occurs 1% of
2.119 A producer of a certain type of electronic component ships to suppliers in lots of twenty. Suppose that 60% of all such lots contain no defective components, 30% contain one defective component, and 10%contain two defective components. A lot is picked, two components from the lot are randomly
2.118 A certain form of cancer is known to be found in women over 60 with probability 0.06. A blood test exists for the detection of the disease, but the test is not infallible. In fact, it is known that 10% of the time, the test gives a false negative (i.e., the test incorrectly gives a negative
2.117 Consider the situation in Exercise 2.116, and suppose that the manufacturer can try only two combinations in a day.(a) What is the probability that any given set of two runs is chosen?(b) What is the probability that the highest temperature is used in either of these two combinations?
2.116 A manufacturer is studying the effects of cooking temperature, cooking time, and type of cooking oil for making potato chips. Three different temperatures, 4 different cooking times, and 3 different oils are to be used.(a) What is the total number of combinations to be studied?(b) How many
2.115 A certain federal agency employs three consulting firms (A, B, and C) with probabilities 0.45, 0.40, and 0.25, respectively. From past experience, it is known that the probability of cost overruns for the firms are 0.05, 0.03, and 0.15, respectively. Suppose a cost overrun is experienced by
2.114 A shipment of 12 televisions contains 3 defective ones. In how many ways can a hotel purchase 5 of these ones and receive at least 2 of the defective ones?
2.113 From a box containing 6 black balls and 3 green balls, 3 balls are drawn in succession, each ball being replaced in the box before the next draw is made. What is the probability that(a) all 3 are the same color?(b) each color is represented?
2.112 From 5 red, 6 green, and 7 yellow apples, how many selections of 9 apples are possible if 3 of each color are to be selected?
2.111 In a certain federal prison, it is known that 2/3 of the inmates are under 25 years of age. It is also known that 3/5 of the inmates are male and that 5/8 of the inmates are female or 25 years of age or older. What is the probability that a prisoner selected at random from this prison is
2.110 The probability that a patient recovers from a delicate heart operation is 0.8. What is the probability that(a) exactly 2 of the next 3 patients who have this operation survive?(b) all of the next 3 patients who have this operation survive?
2.108 If the probability is 0.1 that a person will make a mistake on his or her state income tax return, find the probability that(a) four totally unrelated persons each make a mistake;(b) Mr. Jones and Ms. Clark both make mistakes, and Mr. Roberts and Ms. Williams do not make a mistake.to provide
2.107 How many bridge hands are possible containing 4 spades, 6 diamonds, 1 club, and 2 hearts?
2.106 The probabilities that a service station will pump gas into 0, 1, 2, 3, 4, or 5 or more cars during a certain 30-minute period are 0.03, 0.18, 0.24, 0.28, 0.10, and 0.17, respectively. Find the probability that in this 30-minute period(a) more than 2 cars receive gas;(b) at most 4 cars
2.105 By comparing appropriate regions of Venn diagrams, verify that(a) (A ∩ B) ∪ (A ∩ B) = A;(b) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
2.104 An allergist claims that 50% of the patients she tests are allergic to some type of weed. What is the probability that(a) exactly 3 of her next 4 patients are allergic to weeds?(b) none of her next 4 patients is allergic to weeds?
2.103 A truth serum has the property that 90% of the guilty suspects are properly judged while, of course, 10% of the guilty suspects are improperly found innocent.On the other hand, innocent suspects are misjudged 1% of the time. If the suspect was selected from a group of suspects of which only
2.102 Denote by A, B, and C the events that a grand prize is behind doors A, B, and C, respectively. Suppose you randomly picked a door, say A. The game host opened a door, say B, and showed there was no prize behind it. Now the host offers you the option of either staying at the door that you
2.101 A paint-store chain produces and sells latex and semigloss paint. Based on long-range sales, the probability that a customer will purchase latex paint is 0.75. Of those that purchase latex paint, 60% also purchase rollers. But only 30% of semigloss paint buyers purchase rollers. A randomly
2.100 A regional telephone company operates three identical relay stations at different locations. During a one-year period, the number of malfunctions reported by each station and the causes are shown below.Station A B C Problems with electricity supplied 2 1 1 Computer malfunction 4 3 2
2.99 Suppose that the four inspectors at a chocolate factory are supposed to stamp the expiration date on each package of chocolate at the end of the assembly line. John, who stamps 20% of the packages, fails to stamp the expiration date once in every 200 packages;Tom, who stamps 60% of the
2.98 If the person in Exercise 2.96 received a speeding ticket on her way to work, what is the probability that she passed through the radar trap located at L2 ?
2.97 Referring to Exercise 2.95, what is the probability that a person diagnosed as having cancer actually has the disease?
2.96 Police plan to enforce speed limits by using radar traps at four different locations within the city limits.The radar traps at each of the locations L1 , L2 , L3 , and L4 will be operated 40%, 30%, 20%, and 30% of the time. If a person who is speeding on her way to work has probabilities of
2.95 In a certain region of the country it is known from past experience that the probability of selecting an adult over 40 years of age with cancer is 0.05. If the probability of a doctor correctly diagnosing a person with cancer as having the disease is 0.78 and the probability of incorrectly
2.94 In the situation of Exercise 2.93, it is known that the system does not work. What is the probability that the component A also does not work? 0.7 0.7 A B 0.8 0.8 0.8 D ml E
2.93 A circuit system is given in Figure 2.11. Assume the components fail independently.(a) What is the probability that the entire system works?(b) Given that the system works, what is the probability that the component A is not working?
2.92 Suppose the diagram of an electrical system is as given in Figure 2.10. What is the probability that the system works? Assume the components fail independently.
2.91 Find the probability of randomly selecting 3 good quarts of milk in succession from a cooler containing 20 quarts of which 5 have spoiled, by using(a) the first formula of Theorem 2.12;(b) the formulas of Theorem 2.6 and Rule 2.3, respectively.
2.90 Pollution of the rivers in the United States has been a problem for many years. Consider the following events:A: the river is polluted, B: a sample of water tested detects pollution, C : fishing is permitted.Assume P(A) = 0.3, P(B|A) = 0.75, P(B|A ) = 0.20, P(C|A∩B) = 0.20, P(C|A ∩B) =
2.89 A town has two fire engines operating independently.The probability that a specific engine is available when needed is 0.96.(a) What is the probability that neither is available when needed?(b) What is the probability that a fire engine is available when needed?
2.88 Before the distribution of certain statistical software, every fourth compact disk (CD) is tested for accuracy.The testing process consists of running four independent programs and checking the results. The failure rates for the four testing programs are, respectively, 0.01, 0.03, 0.02, and
2.86 In 1970, 11% of Americans completed four years of college; 43% of them were women. In 1990, 22% of Americans completed four years of college; 53% of them were women (Time, Jan. 19, 1996).(a) Given that a person completed four years of college in 1970, what is the probability that the person
2.85 The probability that a doctor correctly diagnoses a particular illness is 0.8. Given that the doctor makes an incorrect diagnosis, the probability that the patient files a lawsuit is 0.9. What is the probability that the doctor makes an incorrect diagnosis and the patient sues?
2.84 The probability that you are home when a telemarketing representative calls is 0.4. Given that you are home, the probability that goods will be bought from the company is 0.3. Find the probability that you are home and goods are bought from the company.
2.83 The probability that a vehicle entering the Luray Caverns has Canadian license plates is 0.12; the probability that it is a camper is 0.28; and the probability that it is a camper with Canadian license plates is 0.09. What is the probability that(a) a camper entering the Luray Caverns has
2.82 For married couples living in a certain suburb, the probability that the husband will vote on a bond referendum is 0.2, the probability that the wife will vote on the referendum is 0.28, and the probability that both the husband and the wife will vote is 0.15.What is the probability that(a) at
2.81 The probability that a married man watches a certain television show is 0.45, and the probability that a married woman watches the show is 0.5. The probability that a man watches the show, given that his wife does, is 0.8. Find the probability that(a) a married couple watches the show;(b) a
2.80 The probability of an automobile being filled with gasoline also needing an oil change is 0.25; the probability that it needs a new oil filter is 0.40; and the probability that both the oil and the filter need changing is 0.2.(a) If the oil has to be changed, what is the probability that a new
2.79 In USA Today (Sept. 5, 1996), the results of a survey involving the use of sleepwear while traveling were listed as follows:Male Female Total Underwear 0.220 0.024 0.244 Nightgown 0.002 0.180 0.182 Nothing 0.160 0.018 0.178 Pajamas 0.102 0.073 0.175 T-shirt 0.046 0.088 0.134 Other 0.084 0.003
2.78 A manufacturer of a flu vaccine is concerned about the quality of its flu serum. Batches of serum are processed by three different departments having rejection rates of 0.10, 0.05, and 0.15, respectively. The inspections by the three departments are sequential and independent.(a) What is the
2.77 In the senior year of a high school graduating class of 100 students, 42 studied mathematics, 68 studied psychology, 54 studied history, 22 studied both mathematics and history, 25 studied both mathematics and psychology, 7 studied history but neither mathematics nor psychology, 10 studied all
2.76 In an experiment to study the relationship of hypertension and smoking habits, the following data are collected for 180 individuals:Moderate Heavy Nonsmokers Smokers Smokers H 21 36 30 NH 48 26 19 where H and NH in the table stand for Hypertension and Nonhypertension, respectively. If one of
2.75 A random sample of 200 adults are classified below by sex and their level of education attained.Education Male Female Elementary 38 45 Secondary 28 50 College 22 17 If a person is picked at random from this group, find the probability that(a) the person is a male, given that the person has a
2.74 A class in advanced physics is composed of 10 juniors, 30 seniors, and 10 graduate students. The final grades show that 5 of the juniors, 10 of the seniors, and 5 of the graduate students received an A for the course. If a student is chosen at random from this class and is found to have earned
2.73 If R is the event that a convict committed armed robbery and D is the event that the convict pushed dope, state in words what probabilities are expressed by(a) P(R|D);(b) P(D|R);(c) P(R|D).
2.71 As the situation of Exercise 2.69 might suggest, statistical procedures are often used for control of quality(i.e., industrial quality control). At times, the weight of a product is an important variable to control.Specifications are given for the weight of a certain packaged product, and a
2.70 Consider the situation of Exercise 2.69. Suppose 50,000 boxes of detergent are produced per week and suppose also that those underfilled are “sent back,”with customers requesting reimbursement of the purchase price. Suppose also that the cost of production is known to be $4.00 per box
2.69 It is common in many industrial areas to use a filling machine to fill boxes full of product. This occurs in the food industry as well as other areas in which the product is used in the home, for example, detergent. These machines are not perfect, and indeed they may A, fill to specification,
2.68 Interest centers around the nature of an oven purchased at a particular department store. It can be either a gas or an electric oven. Consider the decisions made by six distinct customers.(a) Suppose that the probability is 0.40 that at most two of these individuals purchase an electric
2.67 Consider the situation of Example 2.32 on page 78.(a) What is the probability that no more than 4 cars will be serviced by the mechanic?(b) What is the probability that he will service fewer than 8 cars?(c) What is the probability that he will service either 3 or 4 cars?
2.66 Factory workers are constantly encouraged to practice zero tolerance when it comes to accidents in factories. Accidents can occur because the working environment or conditions themselves are unsafe. On the other hand, accidents can occur due to carelessness or so-called human error. In
2.65 Consider the situation of Exercise 2.64. Let A be the event that the component fails a particular test and B be the event that the component displays strain but does not actually fail. Event A occurs with probability 0.20, and event B occurs with probability 0.35.(a) What is the probability
2.64 Interest centers around the life of an electronic component. Suppose it is known that the probability that the component survives for more than 6000 hours is 0.42. Suppose also that the probability that the component survives no longer than 4000 hours is 0.04.(a) What is the probability that
2.63 The likely location of a mobile device in the home is as follows:Adult bedroom: 0.03 Child bedroom: 0.15 Other bedroom: 0.14 Office or den: 0.40 Other rooms: 0.28(a) What is the probability that a mobile device is in a bedroom?(b) What is the probability that it is not in a bedroom?(c) Suppose
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