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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

Refer to the Journal of Engineering for Industry (May 1993) study of friction feed paper separation, Exercise 5.12. Consider a system that utilizes two interrelated feed paper separators. The joint density of X and Y, the friction coefficients of the two machines, is given bya. Verify that f(x, y)
The joint density of X, the total time (in minutes) between an automobile’s arrival in the service queue and its leaving the system after servicing, and Y, the time (in minutes) the car waits in the queue before being serviced, isa. Find the value of c that makes f (x, y) a probability density
In building construction, a truss is a structure comprised of triangular units whose ends are connected at joints referred to as nodes. The Journal of Engineering Mechanics (Dec. 2009) published a study of the characteristics of a 10-bar truss subjected to loads at four different nodes. Two random
The Department of Transportation (DOT) monitors sealed bids for new road construction. For new access roads in a certain state, let X = low bid (thousands of dollars) and let Y = DOT estimate of fair cost of building the road (thousands of dollars). The joint probability density of X and Y
Let X and Y be two discrete random variables with joint probability distribution p(x, y). DefineF1(a) = P(X ≤ a) and F1 (a Ι y) = P(X ≤ a Ι y)Verify each of the following:a.b. F1(a) = EEP(x, y) Isa y %3D
The joint probability distribution for two discrete random variables, X and Y, is given by the formulaVerify that the properties of a bivariate probability distribution are satisfied. p(x, y) = p*+Yq?-(x+y), x = 0, 1, 0 sps1, q = 1- p, y = 0, 1
The Face Recognition Technology (FERET) program, sponsored by the U.S. Department of Defense, was designed to develop automatic face recognition capabilities to assist homeland security. A biometric face “signature” of an unknown person (called the probe) is compared to a signature of a known
From a group of three data processing managers, two senior systems analysts, and two quality control engineers, three people are to be randomly selected to form a committee that will study the feasibility of adding a dual-core CPU cooler at a consulting firm. Let X denote the number of
A special delivery truck travels from point A to point B and back over the same route each day. There are three traffic lights on this route. Let X be the number of red lights the truck encounters on the way to delivery point B and let Y be the number of red lights the truck encounters on the way
Refer to The International Journal of Robotics Research (Dec. 2004) study of a robot-sensor system in an unknown environment, Exercise 4.7. In the three-point, single-link robotic system shown in the accompanying figure, each point (A, B, or C) in the system has either an “obstacle” status or a
Refer to the Canadian Journal of Civil Engineering (Jan. 2013) investigation of the use of variable speed limits to control freeway traffic congestion, Exercise 4.9. Recall that the study site was an urban freeway divided into three sections with variable speed limits posted in each section. The
Refer to the Engineering Computations: International Journal for Computer-Aided Engineering and Software (Vol. 30, No. 2, 2013) study of the properties of granular media (e.g., sand, rice, ball bearings, and flour), Exercise 3.62. The study assumes there is a system of N non-interacting granular
Refer to the IEEE Transactions on Information Forensics and Security (March 2013) study of wireless identify theft using cloned credit or debit cards, Exercise 3.44. A cloning detection method was illustrated using a simple ball drawing game. Consider the group of 10 balls shown below. Of these, 5
Consider the experiment of tossing a pair of dice. Let X be the outcome (i.e., the number of dots appearing face up) on the first die and let Y be the outcome on the second die.a. Find the joint probability distribution p(x, y).b. Find the marginal probability distributions p1(x) and p2(y).c. Find
A software program is designed to perform two tasks, A and B. Let X represent the number of IF-THEN statements in the code for task A and let Y represent the number of IF-THEN statements in the code for task B. The joint probability distribution p(x, y) for the two discrete random variables is
Let my(t) be the moment generating function of a continuous random variable Y. If a and b are constants, show thata.b.c. my+a(l) = E[e(y+a] = e"m,(1) ealm,(1)
To aid engineers seeking to predict the efficiency of a solar-powered device, Olseth and Skartveit developed a model for daily insolation Y at sea-level locations within the temperate storm belt. To account for both “clear sky” and “overcast” days, the researchers constructed a probability
The continuous random variable Y has a probability distribution given bya. Find the value of c that makes f(y) a probability density.b. Find F(y).c. Compute P(Y > 2.5). (cye if y > 0 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(2.6).d. Show that and F(0) = 0 F(∞) = 1e. Compute P(1 ≤ Y ≤ 5). Sce Y if y > 0 ce-y Ky) elsewhere
In finding and correcting errors in a software code (debugging) and determining the code’s reliability, computer software experts have noted the importance of the distribution of the time until the next coding error is found. Suppose that this random variable has a gamma distribution with
Refer to the Journal of the Acoustical Society of America (Feb. 1986) study of auditory nerve response rates in cats, discussed in Exercise 4.107. A key question addressed by the research is whether rate changes (i.e., changes in number of spikes per burst of noise) produced by tones in the
The life length Y (in years) of a memory chip in a laptop computer is a Weibull random variable with probability densitya. What are the values of α and β?b. Compute the mean and variance of Y.c. Find the probability that a new memory chip will not fail before 6 years. f(y) 1--ve-1/ -/16 if 0 y
The percentage Y of impurities per batch in a certain chemical product is a beta random variable with probability densitya. What are the values of α and β?b. Compute the mean and variance of Y.c. A batch with more than 80% impurities cannot be sold. What is the probability that a randomly
Show that for a Poisson random variable Y, σ2 = λ. ( Use the result of Exercise 4.83 and Theorem 4.4.)Data from Exercise 4.83 Elevator passenger arrivals. A study of the arrival process of people using elevators at a multi-level office building was conducted and the results reported in Build- ing
Verify that the mean of a beta density with parameters α and β is given by μ = α/(α + β).
Prove Theorem 5.1. THEOREM 5.1 Let c be a constant, let Y be a continuous random variable, and let g1(Y), g2(Y), ..., gx(Y) be k functions of Y. Then, E(c) = c E(cY) = cE(Y) E[g (Y) + g2(Y) + + g(Y)] = E[g(Y)] + E[g2(Y)] ++ Elgx(Y)] %3D
Prove Theorem 5.2. THEOREM 5.2 Let Y be a continuous random variable with E(Y) = µ. Then 7 = E[CY – p1 = E(Y?) – p? %3D
Assume that Y is uniformly distributed over the interval 0 ≤ Y ≤ 1. Show that, for a ≥ 0 b, b ≥ 0, and (a + b) ≤ 1,P(a < Y < a + b) = b
Engineers studied the life cycle cost of a coal-fired power plant in the Journal of Quality in Maintenance Engineering (Vol. 19, 2013). The analysis used the Weibull distribution to model the probability distribution of time to failure, Y, measured in thousands of hours. The engineers used β = 65
For each of the following exercises, find μ and σ2. Then compute P(μ – 2σ < Y < μ + 2σ) and compare to the Empirical Rule.a. Exercise 5.1b. Exercise 5.2c. Exercise 5.3d. Exercise 5.4Data From Exercise 5.1:Let c be a constant and consider the density function for the random
Refer to Example 5.15 (p. 197) and the life length Y of a drill bit. Recall that Y has a Weibull distribution with α = 2 and β = 100.a. Calculate the values of f(y) for y = 2, 5, 8, 11, 14 and 17. Plot the points ( y, f(y)) and construct a graph of the failure time distribution of the drill
Japanese electrical engineers have developed a sophisticated radar system called the moving target detector (MTD), designed to reject ground clutter, rain clutter, birds, and other interference. The system was used to successfully detect aircraft embedded in ground clutter. (Scientific and
Based on extensive testing, a manufacturer of washing machines believes that the distribution of the time Y (in years) until a major repair is required has a Weibull distribution with α = 2 and β = 4.a. If the manufacturer guarantees all machines against a major repair for 2 years, what
The length of time (in months after maintenance) until failure of a bank’s surveillance television equipment has a Weibull distribution with α = 2 and β = 60. If the bank wants the probability of a breakdown before the next scheduled maintenance to be .05, how frequently should the equipment
Show that the Weibull distribution with α = 2 and β > 0 is new better than used (NBU).
Show that for the Weibull distribution, a +
Show that for the Weibull distribution,Then use the relationshipto show that EÇY²) = B2/«T( a + 2
Sedimentary rocks are a mixture of minerals and pores. Engineers employ well (or borehole) logging to make a detailed record of the mineral composition of the rock penetrated by a borehole. One important measure is shale density of a rock (measured as a fraction between 0 and 1). In the journal SPE
At the 2006 International Conference on Biomedical Engineering, researchers presented an unsupervised learning technique for segmenting skin images. The method employed various forms of a beta distribution, one of which was symmetric. The parameters of a symmetric beta distribution are equal, i.e.,
In the Journal of Statistical Computation and Simulation (Vol. 67, 2000), researchers presented a numerical procedure for estimating the parameters of a beta distribution. The method involves a re-parameterization of the probability density function. Consider a beta distribution with parameters α*
A block cipher is an encryption algorithm that transforms a fixed-length block of unencrypted text data (called plaintext) into a block of encrypted text data (called ciphertext) of the same length for security purposes. A group of Korean communications engineers have designed a new linear
An investigation into pollution control expenditures of industrial firms found that the annual percentage of plant capacity shutdown attributable to environmental and safety regulation has an approximate beta distribution with α = 1and β = 25.a. Find the mean and variance of the annual percentage
The proportion Y of a data processing company’s yearly hardware repair budget allocated to repair its laser color printer has an approximate beta distribution with parameters α = 2 and β = 9.a. Find the mean and variance of Y.b. Compute the probability that for any randomly selected year, at
Suppose the proportion of Internet start-up firms that make a profit during their first year of operation possesses a relative frequency distribution that can be approximated by the beta density with α = 5 and β = 6.a. Find the probability that at most 60% of all Internet startup firms make a
An important property of certain products that are in powder or granular form is their particle size distribution. For example, refractory cements are adversely affected by too high a proportion of coarse granules, which can lead to weaknesses from poor packing. G. H. Brown (Journal of Quality
A continuous random variable Y has a beta distribution with probability densityFind the value of c that will make f(y) a density function. Scy'(1 - y) if 0 s ys 1 f(y) elsewhere
Show that if Y has a beta density with α = 1and β = 1, then Y is uniformly distributed over the interval 0 ≤ Y ≤ 1.
Show that the beta distribution with α = 2 and β = 1 is new better than used (NBU).
Use the moment generating function m(t) of the normal density to find μ1'and μ2' . Then use these results to show that a normal random variable has mean μ and variance σ2.
Verify that the moment generating function of a chi-square random variable with n degrees of freedom is m(t) = (1 - 2t)-v/2
Consider a continuous random variable Y with densitya. Find the moment generating function m(t) of Y.b. Use the result of part a to find the mean and variance of Y. Se if y < 0 Ry) lo elsewhere
Verify that the moment generating function of a uniform random variable on the interval a ≤ Y ≤ b is elb - eta m(t) t(b – a)
Ecological Applications (May 1995) published a study on the development of forests following wildfires in the Pacific Northwest. One variable of interest to the researcher was tree diameter at breast height 110 years after the fire. The population of Douglas fir trees was shown to have an
A manufacturing company has developed a fuel-efficient machine that combines pressure washing with steam cleaning. It is designed to deliver 7 gallons of cleaner per minute at 1,000 pounds per square inch for pressure washing. In fact, it delivers an amount at random anywhere between 6.5 and 7.5
The problem of passenger congestion prompted a large international airport to install a monorail connecting its main terminal to the three concourses, A, B, and C. The engineers designed the monorail so that the amount of time a passenger at concourse B must wait for a monorail car has a uniform
A pacemaker is made up of several biomedical components that must be of a high quality for the pacemaker to work. It is vitally important for manufacturers of pacemakers to use parts that meet specifications. One particular plastic part, called a connector module, mounts on the top of the
An infestation of a certain species of caterpillar, the spruce budworm, can cause extensive damage to the timberlands of the northern United States. It is known that an outbreak of this type of infestation occurs, on the average, every 30 years. Assuming that this phenomenon obeys an exponential
In an article published in the European Journal of Operational Research, the vehicle-dispatching decisions of an airport based taxi service were investigated. In modeling the system, the authors assumed travel times of successive taxi trips to and from the terminal to be independent exponential
The importance of modeling machine downtime correctly in simulation studies was discussed in Industrial Engineering (Aug. 1990). The paper presented simulation results for a single-machine tool system with the following properties: • The interarrival times of jobs are exponentially
Suppose that the fraction of defective modems shipped by a data-communications vendor has an approximate beta distribution with α = 5 and β = 21.a. Find the mean and variance of the fraction of defective modems per shipment.b. What is the probability that a randomly selected shipment will contain
W. Nelson (Journal of Quality Technology, July 1985) suggests that the Weibull distribution usually provides a better representation for the life length of a product than the exponential distribution. Nelson used a Weibull distribution with α = 1.5 and β = 110 to model the life length Y of a
Digital elevation models (DEM) are now used to estimate elevations and slopes of remote regions. In Arctic, Antarctic, and Alpine Research (May 2004), geographers analyzed reading errors from maps produced by DEM. Two readers of a DEM map of White Glacier (Canada) estimated elevations at 400 points
The Harris Corporation data on voltage readings at two locations, Exercise 2.72, are reproduced at the bottom of the page. Determine whether the voltage readings at each location are approximately normal.Data From Exercise 2.72:A Harris Corporation/University of Florida study was undertaken to
A team of soil scientists investigated the water retention properties of soil cores sampled from an uncropped field consisting of silt loam (Soil Science, Jan. 1995). At a pressure of .1 megapascal (MPa), the water content of the soil (measured in cubic meters of water per cubic meter of soil) was
Refer to the data on percentage iron in 66 bulk specimens of Chilean lumpy iron ore, Exercise 2.79. The data are saved in the LUMPYORE file. Assess whether the data are approximately normal.Data From Exercise 2.79:Lumpy iron ore. Sixty-six bulk specimens of Chilean lumpy iron ore (95%
Suppose we are counting events that occur according to a Poisson distribution, such as the number of automobile accidents at a left-turn lane. If it is known that exactly one such event has occurred in a given interval of time, say (0, t), then the actual time of occurrence is uniformly distributed
Refer to the Journal of Earthquake Engineering study of the time Y (in years) between major earthquakes occurring in the Iranian Plateau, Exercise 5.7. Recall that Y has the following density function:a. Find Interpret the result.b. Find the variance of Y.c. Use the Empirical Rule to estimated.
Refer to Exercises 4.2 and 4.13. Each home with a dust mite level that exceeds 2mg/g will spend $2,000 for an allergen air purification system. Find the mean and variance of the total amount spent by the four sampled homes. Give a range where this total is likely to fall.Data from Exercise 4.2A
Use Theorem 4.4 to calculate the variance of the probability distribution in Exercise 4.4. Interpret the result.Data from Exercise 4.4 The data on the nearshore bar condition for six beach hot spots are reproduced in the table below. Suppose you randomly select two of these six beaches and count Y,
Use Theorem 4.4 to calculate the variance of the probability distribution in Exercise 4.12. Verify that your result agrees with Exercise 4.12.Data from Exercise 4.12According to an August, 2011 survey by the Pew Internet & American Life Project, nearly 40% of adult cell phone owners have
Refer to Exercise 4.11, where Y is the number of firing pins tested in a sample of five selected from a large lot. Suppose the cost of inspecting a single pin is $300 if the pin is defective and $100 if not. Then the total cost C (in dollars) of the inspection is given by the equation C = 200 +
Refer to the Annals of the Entomological Society of America (Jan. 2005) study of the life cycle of a South American delphacid species, Exercise 4.3 (p. 138). Recall that entomological engineers have found that the delphacid is a natural enemy of water hyacinth. The table giving the percentages of
The Environmental Protection Agency (EPA) has established national ambient air quality standards in an effort to control air pollution. Currently, the EPA limit on ozone levels in air is 12 parts per hundred million (pphm). A study examined the long-term trend in daily ozone levels in Houston,
The nuclear industry has made a concerted effort to significantly reduce the number of unplanned rapid emergency shutdowns of a nuclear reactor— called scrams. A decade ago, the mean annual number of unplanned scrams at U.S. nuclear reactor units was four see Exercise 2.81. Assume that the annual
Show that for a Poisson random variable Y, a. 0 s p(y) s 1 b Σρ)-1 y=0 C. E(Y) = A? + A [Hint: First derive the result E[ Y(Y – 1)] = x? from the fact that ELY(Y – 1)] = Ey(y - 1)- y! y=0 (y - 2)! z! Then apply the result E[Y(Y – 1)] = E(Y)² – E(Y).]
Derive the moment generating function of the Poisson random variable Y. WriteThen note that the quantity being summed is a Poisson probability with parameter λet.] m(t) = E(e") = E nd'e %3! y=0 y! (Ae') (de')e- y=0 y! y! y-0
Use the result of Exercise 4.86 to derive the mean and variance of the Poisson distribution.
Researchers at the University of Rochester studied the friction that occurs in the paper feeding process of a photocopier (Journal of Engineering for Industry, May 1993). The coefficient of friction is a proportion that measures the degree of friction between two adjacent sheets of paper in the
Timber beams are widely used in home construction. When the load (measured in pounds) per unit length has a constant value over part of a beam the load is said to be uniformly distributed over that part of the beam. Uniformly distributed beam loads were used to derive the stiffness distribution of
Statistical software packages, such as SAS and MINITAB, are capable of generating random numbers from a uniform distribution. For example, the SAS function RANUNI uses a prime modulus 231 – 1 multiplicative generator with modulus and multiplier 397,204,094 to generate a random variable Y from a
Assume that the random variable Y is uniformly distributed over the interval a ≤ Y ≤ b. Verify the following:a.b. a + b (b – a)? and o? 12
Show that the uniform distribution is new better than used (NBU) over the interval (0, 1).
The Transactions of the ASME (June 2004) presented a model for predicting daily natural gas consumption in urban areas. A key component of the model is the distribution of daily temperatures in the area. Based on daily July temperatures collected in Buenos Aires, Argentina, from 1944 to 2000,
Seismic ground noise describes the persistent vibration of the ground due to surface waves generated from traffic, heavy machinery, winds, ocean waves, and earthquakes. A group of civil engineers investigated the structural damage to a three-story building caused by seismic ground noise in
As part of a risk assessment, quality engineers monitored the corrosion rate (millimeters per year) of a wind turbine system susceptible to corrosion. (Journal of Quality in Maintenance Engineering, Vol. 18, 2013). For demonstration purposes, the corrosion rate Y was modeled as a normal
The alkalinity level of water specimens collected from the Han River in Seoul, Korea, has a mean of 50 milligrams per liter and a standard deviation of 3.2 milligrams per liter. (Environmental Science & Engineering, Sept. 1, 2000.) Assume the distribution of alkalinity levels is approximately
Paleomagnetic studies of Canadian volcanic rock known as the Carmacks Group have recently been completed. The studies revealed that the northward displacement of the rock units has an approximately normal distribution with standard deviation of 500 kilometers (Canadian Journal of Earth Sciences,
Let Y be a normal random variable with mean m and variance σ2. Show thathas mean 0 and variance 1. Y - u Z =
Refer to the 2010 CSI Computer Crime and Security Survey, Exercise 2.13. Recall that the percentage of monetary losses attributable to malicious actions by individuals within the organization (i.e., malicious insider actions) was recorded for 144 firms. The histogram for the data is reproduced
An evaluation of the habitats of endangered salmon species was performed in Conservation Ecology (December 2003). The researchers identified 734 sites (habitats) for Chinook, coho, or steelhead salmon species in Oregon, and assigned a habitat quality score to each. (Scores range from 0 to 36
Foresters periodically “cruise” a forest to determine the size (usually measured as the diameter at breast height) of a certain species of trees. The breast height diameters (in meters) for a sample of 28 trembling aspen trees in British Columbia’s boreal forest are listed here. Determine
Refer to the National Highway Traffic Safety Administration (NHTSA) crash test data for new cars. In Exercise 5.38, you assumed that the driver’s head injury rating is approximately normally distributed. Apply the methods of this chapter to the data saved in the CRASH file to support this
Refer to the data on sanitation scores for 186 cruise ships, first presented in Exercise 2.19. The data are saved in the SHIPSANIT file. Assess whether the sanitation scores are approximately normally distributed.Data from Exercise 2.19To minimize the potential for gastrointestinal disease
Refer to the Minerals Engineering (Vol. 46-47, 2013) study of the impact of calcium and gypsum on the flotation properties of silica in water, Exercise 2.23. Recall that 50 solutions of deionized water were prepared both with and without calcium/ gypsum, and the level of flotation of silica in the
The optimal scheduling of preventative maintenance tests of some (but not all) of n independently operating components was developed in Reliability Engineering and System Safety (Jan., 2006). The time (in hours) between failures of a component was approximated by an exponential distribution with
Refer to the Chance (Summer, 2007) article on phishing attacks at a company, Exercise 2.24. Recall that phishing describes an attempt to extract personal/financial information through fraudulent email. The company set up a publicized email account—called a “fraud
Researchers have discovered that the maximum flood level (in millions of cubic feet per second) over a 4-year period for the Susquehanna River at Harrisburg, Pennsylvania, follows approximately a gamma distribution with α = 3 and β = .07 (Journal of Quality Technology, Jan. 986).a. Find the
Effective maintenance of equipment depends on the ability to accurately forecast the demand for spare parts. The Journal of Quality in Maintenance Engineering (Vol. 18, 2012) developed a statistical approach to forecasting spare parts demand. The methodology used the gamma distribution with
Reaction to tear gas. The length of time Y (in minutes) required to generate a human reaction to tear gas formula A has a gamma distribution with α = 2 and β = 2. The distribution for formula B is also gamma, but with α = 1 and β = 4.a. Find the mean length of time required to generate a human
Show that the variance of a gamma distribution with parameters α and β is αβ2.

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