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engineering
statistics for engineering and the sciences

Statistics For Engineering And The Sciences 6th Edition William M. Mendenhall, Terry L. Sincich - Solutions

Let Y have an exponential distribution with mean β. Show that P (Y > a) = e-α/β.
Refer to the concepts of new better than used (NBU) and new worse than used (NWU) in Theoretical Exercise 5.6. Show that the exponential distribution satisfies both the NBU and NWU properties. (Such a “life” distribution is said to be new same as used or memoryless.)Data from Exercise 5.6Rising
We have stated that a chi-square random variable has a gamma-type density with α = v/2 and β = 2. Find the mean and variance of a chi-square random variable.
Suppose a random variable Y has a probability distribution given byFind the value of c that makes f(y) a density function. AY) = {cy'e-/2 if y > 0 f(Y) = elsewhere
Wind energy is fast becoming a major source of electricity in the U.S. Understanding the distribution of wind speeds is critical in the design of wind farms. Research published in the International Journal of Engineering, Science and Technology (Vol. 3, 2011) established the Weibull distribution as
Fracture toughness of materials. Titanium diboride is an extremely hard ceramic material known for its resistance to mechanical erosion or fracture. The fracture toughness of the material, measured in mega Pascals per meters squared (MPa/m2), was modeled using the Weibull distribution in Quality
Use the moment generating function given in the Key Formulas table at the end of this chapter to derive the mean and variance of a geometric random variable.
Refer to the Process Safety Progress (Dec., 2004) study of industrial accidents caused by management system failures, Exercise 3.78 (p. 127). The table listing the four root causes of system failures (and associated proportions) is reproduced below. Suppose three industrial accidents are randomly
Refer to the IEEE Computer Applications in Power study of an outdoor unmanned watching system designed to detect trespassers, Exercise 3.79. In snowy weather conditions, the system detected 7 out of 10 intruders; thus, the researchers estimated the system’s probability of intruder detection in
Environmental engineers classify consumers into one of five categories (see Exercise 3.77, (p. 127), for a description of each group). The probabilities associated with the groups follow:Let Y equal the number of consumers that must be sampled until the first environmentalist is found. (Note: From
An issue of Consumer Reports found widespread contamination and mislabeling of seafood in supermarkets in New York City and Chicago. The study revealed one alarming statistic: 40% of the swordfish pieces available for sale had a level of mercury above the Food and Drug Administration (FDA) maximum
A waterborne nonbacterial gastroenteritis outbreak occurred in Colorado as a result of a long-standing filter deficiency and malfunction of a sewage treatment plant. A study was conducted to determine whether the incidence of gastrointestinal disease during the epidemic was related to water
An engineering development laboratory conducted an experiment to investigate the life characteristics of a new solar heating panel, designed to have a useful life of at least 5 years with probability. A random sample of 20 such solar panels was selected, and the useful life of each was recorded.a.
Two of the five mechanical engineers employed by the county sanitation department have experience in the design of steam turbine power plants. You have been instructed to choose randomly two of the five engineers to work on a project for a new power plant.a. What is the probability that you will
Lesser-developed countries experiencing rapid population growth often face severe traffic control problems in their large cities. Traffic engineers have determined that elevated rail systems may provide a feasible solution to these traffic woes. Studies indicate that the number of
“Hot spots” are species-rich geographical areas (see Exercise 3.81, p. 127). A Nature (Sept. 1993) study estimated the probability of a bird species in Great Britain inhabiting a butterfly hot spot at .70. Consider a random sample of 4 British bird species selected from a total of 10 tagged
A task force established by the Environmental Protection Agency was scheduled to investigate 20 industrial firms to check for violations of pollution control regulations. However, budget cutbacks have drastically reduced the size of the task force, and they will be able to investigate only 3 of the
The random variable Y, the number of cars that arrive at an intersection during a specified period of time, often possesses (approximately) a Poisson probability distribution. When the mean arrival rate l is known, the Poisson probability distribution can be used to aid a traffic engineer in the
The negative binomial distribution was used to model the distribution of parasites (tapeworms) found in several species of Mediterranean fish (Journal of Fish Biology, Aug. 1990). Assume the event of interest is whether a parasite is found in the digestive tract of brill fish, and let Y be the
A study of natural rock slope movements in the Canadian Rockies over the past 5,000 years revealed that the number of major rockslides per 100 square kilometers had an expected value of 1.57 (Canadian Geotechnical Journal, Nov. 1985).a. Find the mean and standard deviation of Y, the number of major
The manufacturer of a price-reading optical scanner claims that the probability it will misread the price of any product by misreading the “bar code” on a product’s label is .001. At the time one of the scanners was installed in a supermarket, the store manager tested its performance. Let Y
A study of vehicle flow characteristics on acceleration lanes (i.e., merging ramps) at a major freeway in Israel found that one out of every six vehicles uses less than one-third of the acceleration lane before merging into traffic (Journal of Transportation Engineering, Nov. 1985). Suppose we
Refer to Exercise 4.103. Suppose that the number of vehicles using the acceleration lane per minute has a mean equal to 1.1.a. What is the probability that more than two vehicles will use the acceleration lane in the next minute? b. What is the probability that exactly three vehicles will use
Industrial robots are programmed to operate through microprocessors. The probability that one such computerized robot breaks down during any one 8-hour shift is .2. Find the probability that the robot will operate for at most five shifts before breaking down twice.
Benzene, a solvent commonly used to synthesize plastics and found in consumer products such as paint strippers and high-octane unleaded gasoline, has been classified by scientists as a leukemia-causing agent. Let Y be the level (in parts per million) of benzene in the air at a petrochemical plant.
A discharge (or response) rate of auditory nerve fibers [recorded as the number of spikes per 200 milliseconds (ms) of noise burst] is used to measure the effect of acoustic stimuli in the auditory nerve. An empirical study of auditory nerve fiber response rates in cats resulted in a mean of 15
Suppose the random variable Y has a moment generating function given bya. Find the mean of Y.b. Find the variance of Y. 2 31 m(t)
Let Y be a geometric random variable. Show that E(Y) = 1/P . WriteThus,Then use the fact that(The sum of this infinite series is given in most mathematical handbooks.)] E(Y) = PEyq"- where q = 1 - p %3D y=1 and note that dq" dq
The probability generating function P(t) for a discrete random variable Y is defined to beP(t) = E(tY) = po + p1t + p2t2 + .....where pi = P(Y = i).a. Find P(t) for the Poisson distribution. Writeand note that the quantity being summed is a Poisson probability with mean λt.b. Use
Let c be a constant and consider the density function for the random variable Y:a. Find the value of c.b. Find the cumulative distribution function F(y).c. Compute F(1).d. Compute F(.5).e. Compute P(1 ≤ Y ≤ 1.5). Scy if 0 < ys 2 f(y) = elsewhere
Let c be a constant and consider the density function for the random variable Y:a. Find the value of c.b. Find the cumulative distribution function F(y).c. Compute F(.4).d. Compute P(1 ≤ Y ≤ .5). S(2 - y) if 0 sys1 f(y) = elsewhere
Let c be a constant and consider the density function for the random variable Y:a. Find the value of c.b. Find the cumulative distribution function F(y).c. Compute F(–.5).d. Compute P(0 ≤ Y ≤ .5). Sc + y if – 1
Let c be a constant and consider the density function for the random variable Y:a. Find the value of c.b. Find the cumulative distribution function F(y).c. Compute F(1).d. Compute P(Y > .5). (1/c)e y/2 if y 2 0 l1/c)e/2 f(y) if y < 0
The amount of time Y (in minutes) that a commuter train is late is a continuous random variable with probability densitya. Find the value of c for this probability distribution.b. Find the cumulative distribution function, F(y).c. What is the probability that the train is no more than 3 minutes
Rising sea levels are a threat to coastal cities in the United States. Consequently, for planning purposes it is important to have accurate forecasts of the future rise in sea level. The Journal of Waterway, Port, Coastal, and Ocean Engineering (March/April, 2013) published a study which used a
The Journal of Earthquake Engineering modeled the time Y (in years) between major earthquakes occurring in the Iranian Plateau. One of the models considered had the following density function:a. Show that the properties of a density function for a continuous random variable are satisfied for any
Extreme value distributions are used to model values of a continuous random variable that represent extremely rare events. For example, an oceanic engineer may want to model the size of a freak wave from a tsunami, or an environmental engineer might want to model the probability of the hottest
Continuous probability distributions provide theoretical models for the life length of a component (e.g., computer chip, lightbulb, automobile, air-conditioning unit, and so on). Often, it is important to know whether or not it is better to periodically replace an old component with a new
Refer to Exercise 5.5. The amount of time Y (in minutes) that a commuter train is late is a continuous random variable with probability densitya. Find the mean and variance of the amount of time in minutes the train is late.b. Find the mean and variance of the amount of time in hours the train is
Refer to the Journal of Waterway, Port, Coastal, and Ocean Engineering (March/April, 2013) study of coastal sea level rise, Exercise 5.6. Recall that the acceleration Y in sea level rise (standardized between 0 and 1) was modeled using the following density function:ƒ(y) = 6y(1 – y), 0 < y
Refer to Exercise 4.3 and the probability distribution for the number of delphacid eggs on a blade of water hyacinth. Find the mean of the probability distribution and interpret its value.Data from Exercise 4.3Entomological engineers are continually searching for new biological agents to control
Refer to the Journal of Systems Sciences & Systems Engineering (March, 2013) study of the reliability of a manufacturing system that involves multiple production lines, Exercise 4.6. Consider, again, the network for producing integrated circuit (IC) cards with two production lines set up in
The number of training units that must be passed before a complex computer software program is mastered varies from one to five, depending on the student. After much experience, the software manufacturer has determined the probability distribution that describes the fraction of users mastering the
Refer to Exercise 4.11. Suppose the cost of testing a single firing pin is $200.a. What is the expected cost of inspecting the lot?b. What is the variance?c. Within what range would you expect the inspection cost to fall?Data from Exercise 4.11A quality control engineer samples five from a large
Refer to the March, 2013 National Science Foundation study of girls participation in science, technology, engineering or mathematics (STEM) programs, Exercise 1.1 (p.5). Recall that the study found that of girls surveyed who participated in a STEM program, 27% felt that the program increased their
In the Journal of Agricultural, Biological and Environmental Sciences (Sept., 2000), researchers evaluated the risk posed by hazardous pollutants using an ecotoxicological survival model. For one experiment, 20 guppies (all the same age and size) were released into a tank of natural seawater
The National Science Foundation reports that 70% of the U.S. graduate students who earn PhD degrees in engineering are foreign nationals. Consider the number Y of foreign students in a random sample of 25 engineering students who recently earned their PhD.a. Find P(Y = 10).b. Find P(Y ≤ 5).c.
Refer to the Engineering Applications of Artificial Intelligence (Vol. 26, 2013) study of the confidence level of feedback information generated by a semiconductor, Exercise 3.41. Recall that at any point in time during the production process, a report can be generated indicating the system is
A quality control engineer samples five from a large lot of manufactured firing pins and checks for defects. Unknown to the inspector, three of the five sampled firing pins are defective. The engineer will test the five pins in a randomly selected order until a defective is observed (in which case
Exercise 4.2 gives the probability distribution for the number Y of homes with high dust mite levels.a. Find E(Y). Give a meaningful interpretation of the result. b. Find s.c. Find the exact probability that Y = y is in the interval μ ± 2σ. Compare to Chebyshev’s Rule and the Empirical
Refer to Example 4.6. The probability distribution for the time to evacuate in the event of a hurricane, Table 4.4, is reproduced here. Weather forecasters say they cannot accurately predict a hurricane landfall more than 14 hours in advance. If the Gulf Coast Civil Engineering Department waits
Refer to the neutral particle transport problem described in Exercise 3.42 (p. 84). Recall that particles released into an evacuated duct collide with the inner duct wall and are either scattered (reflected) with probability .16 or absorbed with probability .84.a. If 4 particles are released into
Ataxia-telangiectasia (A-T) is a neurological disorder that weakens immune systems and causes premature aging. Science News reports that when both members of a couple carry the A-T gene, their children have a 1 in 5 chance of developing the disease.a. Consider 15 couples in which both members of
For the binomial probability distribution p(y), show that ∑ny p(y) = 1 The binomial theorem, which pertains to the expansion of (a + b)n, states thatLet a = q and b = p.] (a + by - (6)~ - ()- () a"-b +
Show that, for a binomial random variable, E[Y(Y - 1)] = NPQ + μ2 - μ. write the expected value as a sum, factor out y(y - 1), and then factor terms until each term in the sum is a binomial probability. Use the fact that ∑y p(y) = 1 to sum the series.
Refer to the Computers & Security (July, 2013) study of Microsoft program security issues, Exercise 2.4. Recall that Microsoft periodically issues a Security Bulletin that reports the software -- Windows, Explorer, or Office — affected by the vulnerability. The study discovered that 64% of
Dust explosions in the chemical process industry, although rare, pose a great potential for injury and equipment damage. Process Safety Progress (Sept., 2004) reported on the likelihood of dust explosion incidents. The table gives the proportion of incidents in each of several worldwide industries
Refer to the Journal of Transportation Engineering (May, 2013) investigation of the assignment of tracks to trains at a busy railroad station, Exercise 2.8 (p. 28). Ideally, engineers will assign trains to tracks in order to minimize waiting time and bottlenecks. Assume there are 10 tracks at the
To compensate for disorientation in zero gravity, astronauts rely heavily on visual information to establish a top-down orientation. The potential of using color brightness as a body orientation clue was studied in Human Factors (Dec. 1988). Ninety college students, reclining on their backs in the
Although illegal, overloading is common in the trucking industry. Minnesota Department of Transportation (MDOT) engineers monitored the movements of overweight trucks on an interstate highway using an unmanned, computerized scale that is built into the highway. Unknown to the truckers, the scale
A sample of size n is selected from a large lot of shear drill bits. Suppose that a proportion p1 contains exactly one defect and a proportion p2 contains more than one defect (with p1 + p2 < 1 ). The cost of replacing or repairing the defective drill bits is C = 4Y1 + Y2 , where Y1 denotes
An electrical current traveling through a resistor may take one of three different paths, with probabilities p1 = .25, p2 = .30 and p3 = .45, respectively. Suppose we monitor the path taken in n = 10consecutive trials.a. Find the probability that the electrical current will travel the first path Y1
For a multinomial distribution with and , verify that [Hint: Use the binomial theorem (see Theoretical Exercise 4.38) to expand the sum [a + (b + c)]2 , then substitute the binomial expansion of (b + c)2 in the resulting expression. Finally, substitute a = p1, b = p2, and c = p3.]
The distributional pattern of pulmonated slugs inhabiting Libya was studied in the AIUB Journal of Science and Engineering (Aug. 2003). The number of slugs of a certain species found in the survey area was modeled using the negative binomial distribution. Assume that the probability of observing a
Refer to the Image Power Green Brands Survey of international consumers, Exercise 3.4 (p. 84). Recall that a “green” product is one built from recycled materials that has minimal impact on the environment. The reasons why a consumer identifies a product as green are summarized in the next
A drought is a period of abnormal dry weather that causes a serious hydrologic imbalance in an area. The Palmer Drought Index (PDI) is designed to quantitatively measure the severity of a drought. A PDI value of -1 or less indicates a dry (drought) period. A PDI value greater than -1 indicates a
The carbon-nitrogen oxygen (CNO) cycle is one type of fusion reaction by which stars convert hydrogen to helium. The distribution of high energy charges resulting from proton–CNO interaction in space was investigated in the Journal of Physics G: Nuclear and Particle Physics (Nov. 1996). When a
Prior to the grounding of space shuttle program, the National Aeronautics and Space Administration (NASA) estimated that the chance of a “critical- item” failure within a space shuttle’s main engine was approximately 1 in 63. The failure of a critical item during flight would lead directly to
Refer to the Nuclear Science and Engineering study, Exercise 4.36. If neutral particles are released one at a time into the evacuated duct, find the probability that more than five particles will need to be released until we observe two particles reflected by the inner duct wall.Data from Exercise
Assume that hitting oil at one drilling location is independent of another, and that, in a particular region, the probability of success at an individual location is .3.a. What is the probability that a driller will hit oil on or before the third drilling?b. If Y is the number of drillings until
Let Y be a negative binomial random variable with parameters r and p. Then it can be shown that W = Y - r is also a negative binomial random variable, where W represents the number of failures before the rth success is observed. Use the facts that E(Y) rq and o to show that E(W) rq and o
Refer to the International Conference on Social Robotics (Vol. 6414, 2010) study of the trend in the design of social robots, Exercise 3.1 (p. 84). The study found that of 106 social robots, 63 were built with legs only, 20 with wheels only, 8 with both legs and wheels, and 15 with neither legs nor
Refer to the Evolutionary Ecology Research (July 2003) study of the patterns of extinction in the New Zealand bird population, Exercise 3.28. Of the 132 bird species saved in the NZBIRDS file, 38 are extinct. Suppose you randomly select 10 of the 132 bird species (without replacement) and record
An article in The American Statistician (May 1991) described the use of probability in a reverse cocaine sting. Police in a midsize Florida city seized 496 foil packets in a cocaine bust. To convict the drug traffickers, police had to prove that the packets contained genuine cocaine. Consequently,
Show that the mean of a hypergeometric random variable Y is μ = nr/N. Show that and then use the fact thatis the hypergeometric probability distribution for Z = (Y - 1), where Z is the number of S’s in (n - 1) trials, with a total of (r - 1) S’s in (N - 1) elements.] (N 1-(r-1)) - 1- (у
U.S. airlines average about 1.6 fatalities per month. (Statistical Abstract of the United States: 2010.) Assume the probability distribution for Y, the number of fatalities per month, can be approximated by a Poisson probability distribution.a. What is the probability that no fatalities will occur
In the Journal of Transportation Engineering (June 2005), the number of non-home-based trips per day taken by drivers in Korea was modeled using the Poisson distribution with λ = 1.15.a. What is the probability that a randomly selected Korean driver will take no more than two non-home-based trips
Penumbral imaging is a technique used by nuclear engineers for imaging objects (e.g., X-rays and lasers) that emit high-energy photons. In IEICE Transactions on Information & Systems (Apr. 2005), researchers demonstrated that penumbral images are always degraded by noise, where the number Y of
Engineers at the University of New Mexico modeled the number of casualties (deaths or missing persons) experienced by a deep-draft U.S. flag vessel over a 3-year period as a Poisson random variable, Y. The researchers estimated E(Y) to be .03. (Management Science, Jan. 1999.)a. Find the variance of
Refer to the U.S. Army Corps of Engineers study of beach erosional hot spots, Exercise 2.5. The data on the nearshore bar condition for six beach hot spots are reproduced in the table. Suppose you randomly select two of these six beaches and count Y, the total number in the sample with a planar
Consider the segment of an electric circuit with three relays shown here. Current will flow from A to B if there is at least one closed path when the switch is thrown. Each of the three relays has an equally likely chance of remaining open or closed when the switch is thrown. Let Y represent the
In high-volume machining centers, cutting tools are replaced at regular, heuristically chosen intervals. These intervals are generally untimely, i.e., either the tool is replaced too early or too late. The Journal of Engineering for Industry (Aug. 1993) reported on an automated
Refer to Example 3.19. (p. 109).a. Find the probability that an intruder is detected, given a clear dayb. Find the probability that an intruder is detected, given a cloudy day.Data from Example 3.19An unmanned monitoring system uses high-tech video equipment and microprocessors to detect intruders.
Based on a study of acid rain, the National Acid Precipitation Assessment Program (NAPAP) estimates the probability at .14 of an Adirondack lake being acidic. Given that the Adirondack lake is acidic, the probability that the lake comes naturally by its acidity is .25 (Science News, Sept. 15,
At the beginning of World War II, a group of British engineers and statisticians was formed in London to investigate the problem of the lethality of antiaircraft weapons.* One of the main goals of the research team was to assess the probability that a single shell would destroy (or cripple)
Since 1961, parcels of land that may contain oil have been placed in a lottery, with the winner receiving leasing rights (at $1 per acre per year) for a period of 10 years. United States citizens 21 years or older are eligible and are entitled to one entry per lottery by paying a $10 filing fee to
Experience has shown that a manufacturer of rewritable CDs produces, on the average, only 1 defective CD in 100. Suppose that of the next 4 CDs manufactured, at least 1 is defective. What would you infer about the claimed defective rate of .01? Explain.
A security alarm system is activated and deactivated by correctly entering the appropriate three-digit numerical code in the proper sequence on a digital panel.a. Compute the total number of possible code combinations if no digit may be used twice.b. Compute the total number of possible code
A full-scale reinforced concrete building was designed and tested under simulated earthquake loading conditions (Journal of Structural Engineering, Jan. 1986). After completion of the experiments, several design engineers were administered a questionnaire in which they were asked to evaluate two
Can man communicate with a machine through brain-wave processing? This question was the topic of research reported in IEEE Engineering in Medicine and Biology Magazine (Mar. 1990). Volunteers were wired to both a computer and an electroencephalogram (EEG) monitor. Each subject performed five tasks
Engineers have a team for unaided human acts of lifting, lowering, pushing, pulling, carrying, or holding and releasing an object—manual materials handling activities (MMHA). M. M. Ayoub, et al. (1980) have attempted to develop strength and capacity guidelines for MMHA. The authors point out that
Writing a C++ program.a. A professor asks his class to write a C++ computer program that prints all three-letter sequences involving the five letters A, B, E, T, and O. How many different three-letter sequences will need to be printed? b. Answer part a if the program is to be modified so that
Five construction companies each offer bids on three distinct Department of Transportation (DOT) contracts. A particular company will be awarded at most one DOT contract.a. How many different ways can the bids be awarded?b. Under the assumption that the simple events are equally likely, find the
An assembler of computer routers and modems uses parts from two sources. Company A supplies 80% of the parts and company B supplies the remaining 20% of the parts. From past experience, the assembler knows that 5% of the parts supplied by company A are defective and 3% of the parts supplied by
To ascertain the effectiveness of their advertising campaigns, firms frequently conduct telephone interviews with consumers using random-digit dialing. With this method, a random number generator mechanically creates the sample of phone numbers to be called.a. Explain how the random number table
To ensure delivery of its raw materials, a company has decided to establish a pattern of purchases with at least two potential suppliers. If five suppliers are available, how many choices (options) are available to the company?
A manufacturing operation utilizes two production lines to assemble electronic fuses. Both lines produce fuses at the same rate and generally produce 2.5% defective fuses. However, production line 1 recently suffered mechanical difficulty and produced 6.0% defectives during a 3-week period. This
Recently, the National Aeronautics and Space Administration (NASA) purchased a new solar-powered battery guaranteed to have a failure rate of only 1 in 20. A new system to be used in a space vehicle operates on one of these batteries. To increase the reliability of the system, NASA installed three
A brewery utilizes two bottling machines, but they do not operate simultaneously. The second machine acts as a backup system to the first machine and operates only when the first breaks down during operating hours. The probability that the first machine breaks down during operating hours is .20.
A company specializing in data-communications hardware markets a computing system with two types of hard disk drives, four types of display stations, and two types of interfacing. How many systems would the company have to distribute if it received one order for each possible combination of hard

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