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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 bottle manufacturing process, Exercise 16.5. To monitor the process mean, three finished bottles are sampled from the production process at 20 points in time (days). The data (weight, in ounces) for last month’s inspection are provided in the table. Construct both an R-chart and
B. Render (Rollins College) and R. M. Stair (Florida State University) presented the case of the Bayfield Mud Company (Quantitative Analysis of Management, 1997). Bayfield supplies boxcars of 50-pound bags of mud-treating agents to the Wet-Land Drilling Company. Mud-treating agents are used to
For a lot of 250 electron tubes with an acceptance quality level of 10%, find the appropriate MIL-STD-105D general inspection sampling plan under each of the following inspection categories:a. Normal inspection levelb. Tightened inspection level
High-level computer technology has developed bit-sized microprocessors for use in operating industrial “robots.” To monitor the fraction of defective microprocessors produced by a manufacturing process, 50 microprocessors are sampled each hour. The results for 20 hours of sampling are provided
A construction engineer buys steel cable in large rolls to use in supporting equipment and temporary structures during the process of erecting permanent structures. Specifications require the breaking strength of the steel cable to exceed 200 pounds. For a lot size of 1,500 large rolls of steel
Refer to the stress analysis on epoxy-repaired truss joints described in Exercise 7.20. Tests were conducted on epoxy-bonded truss joints made of wood to determine tolerances for actual glue line shear stress (Journal of Structural Engineering, Feb. 1986). The mean and standard deviation of the
Consider the airline check-in process described in Exercise 16.76.a. Assume the process is under control and construct a capability analysis diagram for the process. Management has specified an upper specification limit of 5 minutes.b. Is the process capable? Justify your answer.c. If it is
A manufacturer of golf clubs has received numerous complaints about the performance of its graphite shafts. To monitor the shaft production process, a pultrusion method was used. A fabric is pulled through a thermosetting polymer bath and then through a long, heated steel die. As it moves through
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 number of failures per hour of a component was approximated by a Poisson distribution with mean λ.
Suppose the failure time distribution for a product can be approximated by a normal distribution with μ = 3 and σ = 1. a. Find f(t), F(t), and z(t) for t = 0, 1, 2, . . . , 6. b. Plot the values of z(t) for corresponding values of t and obtain a graph of the hazard rate for this normal
The lifetime T (in hours) of a certain electronic component is a random variable with density functiona. Find F(t) and R(t).b. What is the reliability of the component at t = 25 hours?c. Find z(t) and interpret the result. e-/100 t>0 100 f(t) elsewhere
An article published in the annual Journal of Industrial Engineering (2013) studied preventive maintenance operations for fuel dispensers in a chain of gas stations. The time between failures of fuel dispensers at high failure stations was assumed to have an exponential distribution with a mean of
The reliability of an electrical power system in Iraq was analyzed in the journal PLoS ONE (Aug. 2013). The failure time distribution, f(t), followed a Dagum probability distribution with parameters β, α, and k. The density function takes the form:The researchers showed that the reliability
A drill bit has a failure time distribution given by the densitya. Find F(t).b. Find expressions for the reliability R(t) and the hazard rate z(t) of the drill bit at time t.c. Use the results of part b to find R(8) and z(8). 2te/100 0 st
Mechanical engineers at Purdue University conducted a reliability analysis of lead-free solder joints used in microscopic electronic packages (Electronic Components and Technology Conference, May 2005). The minimum time required by cracks in the solder joints to propagate through the weakest joint
Solder joints are widely used in the electronic packaging industry to provide a connection between wafer level chips and a printed circuit board (PCB). The most common failure of the connection is caused by cracks in the bulk solder joint close to the wafer package-pad. Researchers at National
The reliability of wafer level- chip-scale packages mounted on a printed circuit board (PCB) in handheld devices such as cellular phones, pagers, and PDAs was investigated by researchers at National Semiconductor Corporation (Electronic Components and Technology Conference, May 2005). In one
A wet-mix, steel-fiber-reinforced microsilica concrete (called shotcrete), used extensively in Scandinavia, is now being marketed in the United States. The material is said to have a minimum 28-day breaking strength of 9,000 pounds per square inch (psi) of compression. To investigate the breaking
Failure times for two machines at the Babel tire factory in Iraq were obtained and used to estimate mean failure time. (Iraqi Journal of Statistical Science, Vol. 14, 2008.) The data (in hours) for the cutting layers machine and the coating machine are listed in the accompanying table. Failure time
Suppose that an integrated circuit chip possesses an exponential failure time distribution. Fifteen chips were put on accelerated life test until five of the chips failed. The first five failures occurred at 18.2, 19.5, 24.8, 31.0, and 45.6 (in thousands of hours).a. Find a 95% confidence interval
A sample of 100 high-reliability capacitors was placed on test for 2,000 hours. At the end of this period only three capacitors had malfunctioned, with failure times of 810, 1,422, and 1,816 hours. Assuming an exponential failure time distribution, construct a 99% confidence interval for the mean
Refer to Exercise 17.15.a. Find a 99% confidence interval for the hazard rate of the capacitors.b. Find a 99% confidence interval for the reliability of the capacitors at 3,000 hours.c. Find a 99% confidence interval for the probability that a capacitor will fail before 2,000 hours.Data from
A study was conducted to estimate the mean life (in miles) of a certain type of locomotive using censored sampling (Technometrics, May 1985). Ninety-six locomotives were operated for either 135 thousand miles or until failure. Of these, 37 failed before the 135- thousand-mile period. The
The Weibull probability distribution was used to model the survival time of brain cancer patients at an atomic medicine and radiance hospital (Journal of Basra Research Science, Vol. 37, 2011). The data in the table represent the number of 50 brain cancer patients surviving each month, for 10
Suppose the life length (in years) of a memory chip in a mainframe computer has a Weibull failure time distribution. To estimate the Weibull parameters, α and β, 50 chips were placed on test and the number of survivors was recorded at the end of each year, for a period of 8 years. The data are
Refer to Exercise 17.19.a. Use the estimates of α and β to find the probability that a memory chip will fail before 5 years.b. Estimate the reliability of the memory chips at time t = 7 years.Data from Exercise 17.19Suppose the life length (in years) of a memory chip in a mainframe computer has a
Refer to Exercise 17.19.a. Using the least-squares estimates of α and β, find and graph the hazard rate, z(t).b. Compute the hazard rate at time t = 4 years and interpret its value.Data from Exercise 17.19Suppose the life length (in years) of a memory chip in a mainframe computer has a Weibull
Engineers often use a Weibull failure time distribution for a “weakest link” product, i.e., a product consisting of multiple parts (e.g., roller bearings) that fails when the first part (or weakest link) fails. Nelson (Journal of Quality Technology, July 1985) applied the Weibull distribution
A manufacturer of washing machines conducted a life test during which it monitored 12 new machines for a period of 3 years and recorded the time to a major repair for each. At the end of the 3-year testing period, two machines had not yet required a major repair. The failure times (in months) of
To evaluate the performance of rebuilt hydraulic pumps at an aircraft rework facility, 20 pumps were placed on test and the number of pumps still running at the end of each week was recorded for a period of 6 weeks, as listed in the accompanying table.a. Use the data to estimate the parameters, α
Consider a series system consisting of four components, A, B, C, and D, with probabilities of functioning given by pA = .88, pB = .95 pC = .90, and pD = .80. Find the reliability of the system.
Consider a parallel system consisting of four components, A, B, C, and D, with probabilities of functioning given by pA = .90, pB = .99, pC = .92 and pD = .85. Find the reliability of the system.
The reliability of a system consisting of three identical components is .95. What must be the probability of functioning for each component if:a. The components are connected in series?b. The components are connected in parallel?
A certain component has an exponential failure time distribution with mean β = 3 hours.a. Find the probability that the component will fail before time t = 2 hours.b. What is the reliability of the component at time t = 5 hours? Interpret this value.c. Find and graph the hazard rate for the
Suppose the life length (in hours) of a fluorescent light has a Weibull failure time distribution with parameters α = .05 and β = .70.a. Find the probability that the fluorescent light will fail before time t = 1,000 hours.b. Find the reliability of the fluorescent light at time t = 500 hours and
To investigate the performance of the central processing unit (CPU) of a certain type of microcomputer, 20 CPUs were placed on test for a period of 5,000 hours. When the test was terminated, four CPUs had failed with failure times of 1,850, 2,090, 3,440, and 3,970 hours. Assume a negative
Four components, A, B, C, and D, are connected in parallel. Suppose that components A and B have normal failure time distributions with parameters μ = 500 hours and σ = 100 hours, whereas components C and D have Weibull failure time distributions with parameters α = .5 and β = 100. Find the
The service life (in hours) of a semiconductor has an approximate exponential failure time distribution. Ten semiconductors are placed on life test until four fail. The failure times for these four semiconductors are 585, 972, 1,460, and 2,266 hours.a. Construct a 95% confidence interval for the
In Reliability Engineering and System Safety (Jan. 2006), nuclear and quantum engineers at the Korea Advanced Institute of Science and Technology designed a digital safety system for testing system software. Consider k independent software statements in system software code, each with probability
The Center for High Assurance Computer Systems at the Naval Research Laboratory in Washington, D.C. has developed several theoretical models for detecting intruder attacks on high-consequence computer systems (International Information Assurance Workshop, March 2005). Consider a naive attacker
A certain type of coating for pipes is designed to resist corrosion. Five hundred pieces of coated pipe were placed on test and subjected to a 90% solution of hydrochloric acid. At the end of each hour, for a period of 5 hours, the number of pipe specimens that had resisted corrosion was recorded,
A piece of equipment consists of seven tubes connected as shown in the diagram below. Find the reliability of the system if the tubes have probabilities of functioning given by pA = .80, pB = .90, pC = .85, pD = .85, pE = .75, pF = .75, and pG = .95. D A. B E F
A system consists of eight components, as shown in the accompanying diagram. Find the reliability of the system, given that the individual probabilities of functioning are pA = .90, pB = .95, pC = .85, pD = .85, pE = .98, pF = .80, pG = .95, and pH = .95. B A D F E G H
Consider an electrical circuit consisting of two sub circuits, the first of which involves components A, B, and C in parallel and the second of which involves components D and E in parallel. Suppose that the individual reliabilities of the components are given by pA = .95, pB = .95, pC = .90, pD =
Consider the gamma failure time distribution with α = 2 and β = 1 given by the density functiona. Find F(t). ∫te-t dt = -te-t + ∫e-t dtb. Find expressions for the reliability R(t) and the hazard rate z(t).c. Use the results of part b to find R(3) and z(3). Interpret these values. |te 0st
Consider the uniform failure time distribution given by the density functiona. Find F(t), R(t), and z(t).b. Graph the hazard rate z(t) for t = 0, 1, 2, . . . , 5 when β = 10.c. Compute the reliability of the system at t = 4 when β = 10. f(1) 0sts B B elsewhere
Two resistors connected in series have exponential failure time distributions with mean β = 1,000 hours. At time t = 1,000 hours, what is the reliability of the system?
Consider the product system shown in the diagram. Given the individual component reliabilities pA = .85, pB = .75, pC = .75, pD = .90, and pE = .95, find the overall reliability of the system. A B D E
Show that the hazard rate z(t) can be expressed asMake use of the fact that R(t) = 1 - F(t) and, hence, thatThen substitute these expressions into the formula given in Definition 17.5. -d[In R(t)] z(1) dt
The failure times (in hours) of electronic components in a guidance system for a missile have a Weibull distribution with unknown parameters α and β. To derive estimates of these parameters, 1,000 components were placed on life test and every 50 hours the number of components still in operation
Use the result of Exercise 17.44 and the relation f(t) = z(t) R(t) to show that the failure time density can be expressed asThe differential equationhasas its solution. At) = z(t)e/otv) dy f(1)
Suppose we are concerned only with the initial failure of a component. That is, once the component has survived past a certain time t = α, we treat the component (for all practical purposes) as if it never failed. In this situation it is reasonable to use the hazard ratea. Use the result of
Let y1, y2, . . . , yn be a random sample from a gamma distribution with parameters α = 2 and β unknown.a. Show that y̅ is a biased estimator of β. Compute the bias.b. Show that β̂ = y̅/2 is an unbiased estimator of β.c. Find the variance of β̂ = y̅/2. Recall that, for a gamma
Let y1 be a sample of size 1 from a uniform distribution over the interval from 2 to θ.a. Show that y1 is a biased estimator of θ and compute the bias.b. Show that 2(y1 - 1) is an unbiased estimator of θ.c. Find the variance of 2(y1 - 1).
Japanese researchers have developed a compression/depression method of testing electronic circuits based on Huffman coding. (IEICE Transactions on Information & Systems, Jan. 2005.) The new method is designed to reduce the time required for input decompression and output compression—called
Infrasound refers to sound waves or vibrations with a frequency below the audibility range of the human ear. Even though infrasound cannot be heard, it can produce physiological effects in humans. Mechanical science engineers in China studied the impact of infrasound on a person’s blood pressure
Each year the National Highway Traffic Safety Administration (NHTSA) conducts crash tests for new cars. Crash-test dummies are placed in the driver’s seat and front passenger’s seat of a new-car model, and the car is steered by remote control into a head-on collision with a fixed barrier while
Refer to the Environmental & Engineering Geoscience (Nov. 2012) study of settlement of shallow foundations on cohesive soil, Exercise 7.50. Recall that the researchers sampled 13 structures built on a shallow foundation and measured both the actual settlement value (measured in millimeters) and
Refer to the Computers & Security (July 2013) study of security issues with Microsoft products, Exercise 2.4. Recall that Microsoft periodically issues a Security Bulletin that reports the software affected by the vulnerability. In a sample of 50 bulletins issued in a recent year, 32 reported a
Environmental Science & Technology (Jan. 2005) reported on a study of the reliability of a commercial kit to test for arsenic in groundwater. The field kit was used to test a sample of 328 groundwater wells in Bangladesh. If the color indicator on the field kit registers red, the level of
Refer to the CSI Computer Crime and Security Survey, first presented in Exercise 1.19. Recall that of the 351 organizations that responded to the survey, 144 (or 41%) admitted unauthorized use of computer systems at their firms during the year. Estimate the probability of unauthorized use of
Refer to the American Journal of Science (Jan. 2005) study of the chemical makeup of buried glacial drifts (or tills) in Wisconsin, Exercise 2.22. The ratio of aluminum (Al) to beryllium (Be) in sediment for each of a sample of 26 buried till specimens is given in the table at the bottom of the
Astronauts often report episodes of disorientation as they move around the zero gravity spacecraft. To compensate, crew members rely heavily on visual information to establish a top-down orientation. An empirical study was conducted to assess the potential of using color brightness as a body
Refer to the Environmental Science & Technology (Jan. 2005) study of methyl tert-butyl ether (MTBE) contamination in New Hampshire wells, Exercise 2.12 (p. 29). Data collected for a sample of 223 wells are saved in the MTBE file. Recall that each well was classified according to well class
In sealed bidding on state road construction contracts, the “winner’s curse” is the phenomenon of the winning (or highest) bid price being above the expected price of the contract (called the Department of Transportation engineer’s estimate). The Review of Economics and Statistics (Aug.
Erythropoietin (EPO) is a banned drug used by athletes to increase the oxygen-carrying capacity of their blood. New tests for EPO were first introduced prior to the 2000 Olympic Games held in Sydney, Australia. Chance (Spring 2004) reported that of a sample of 830 world-class athletes, 159 did not
For each of the following combinations of a and degrees of freedom (df), find the value of chi-square, xα2 , that places an area α in the upper tail of the chi-square distribution:a. α = .05, df = 7b. α = .10, df = 16c. α = .01, df = 10d. α = .025, df = 8e. α = .005, df = 5
Finding F values. Find Fα for an F distribution with 15 numerator df and 12 denominator df for the following values of α:a. α = .025 b. α = .05 c. α = .10
Refer to the International Journal of Foresting Engineering (Vol. 19, 2008) investigation of the muscle activity patterns in the neck and upper extremities among forestry vehicle operators, Exercise 7.38. Recall that two types of harvesting vehicles — Timber jack and Valmet — were compared
Refer to the Journal of Aqua cultural Engineering (Nov. 2012) study of the feasibility of laser scanning to estimate the volume of fish in a tank, Exercise 7.25. Recall that turbot fish were reared in a tank for experimental purposes and laser scans were executed in randomly selected locations in
The characteristics of sweet potato chips fried at different temperatures were investigated in the Journal of Food Engineering (Sep., 2013). A sample of 6 sweet potato slices were fried at 130º using a vacuum fryer. One characteristic of interest to the researchers was internal oil content
Finding F values. Find F.05 for an F distribution with:a. Numerator df = 7, denominator df = 25b. Numerator df = 10, denominator df = 8 c. Numerator df = 30, denominator df = 60d. Numerator df =15, denominator df = 4
Show that E[θ̂ - θ)2] = V(θ̂ ) + b2(θ)where the bias b(θ) = E(θ̂ ) - (θ). Write (θ̂ - θ) = [θ̂ - E(θ̂ )] + [E(θ̂ ) - θ].
Let y1, y2, . . . , yn be a random sample from a normal distribution, with mean μ and variance σ2. Show that the variance of the sampling distribution of s2 is 2σ4/(n - 1).
A binomial experiment consisting of n trials resulted in Bernoulli observations y1, y2, . . . , yn, whereand P(yi = 1) = p, P(yi = 0) = 1 - p. Let Y = ∑ni=1 yi be the number of successes in n trials.a. Find the moment estimator of p.b. Is the moment estimator unbiased?c. Find the
Let y1, y2, . . . , yn be a random sample of n observations from a Poisson distribution with probability functiona. Find the maximum likelihood estimator of λ.b. Is the maximum likelihood estimator unbiased? P(y) (y = 0, 1, 2,...) %3D y!
Let y1, y2, . . . , yn be a random sample of n observations on a random variable Y, where f(y) is a gamma density function with α = 2 and unknown β:a. Find the maximum likelihood estimator of β.b. Find E(β̂ ) and V(β̂ ). Ve /B if y >0 Ay) = B? otherwise
Refer to Exercise 7.10.a. Find the moment estimator of β.b. Find E(β̂ ) and V(β̂ ).Data from Exercise 7.10Let y1, y2, . . . , yn be a random sample of n observations on a random variable Y, where f(y) is a gamma density function with α = 2 and unknown β: Ve /B if y > 0 fy) = B? otherwise
Let y1, y2, . . . , yn be a random sample of n observations from a normal distribution with mean 0 and unknown variance σ2. Find the maximum likelihood estimator of σ2.
Let y1, y2, . . . , yn be a random sample of n observations from an exponential distribution with densitya. Find the moment estimator of β.b. Is the moment estimator unbiased?c. Find V(β̂ ). y/B if y> 0 f(y) = otherwise
Use Table 7 of Appendix B to determine the values of tα/2 that would be used in the construction of a confidence interval for a population mean for each of the following combinations of confidence coefficient and sample size:a. Confidence coefficient .99, n = 18b. Confidence coefficient .95, n =
It can be shown (proof omitted) that as the sample size n increases, the T distribution tends to normality and the value tα, such that P(T > tα) = α, approaches the value zα, such that P(Z > zα) = α. Use Table 7 of Appendix B to verify that as the sample size n gets infinitely large,
Let Y be the number of successes in a binomial experiment with n trials and probability of success p. Assuming that n is large, use the sample proportion of successes p̂ = Y/n to form a confidence interval for p with confidence coefficient (1 - α). Start with the pivotal statistic and use
Let y1, y2, . . . , yn be a random sample from a Poisson distribution with mean λ. Suppose we use y̅ as an estimator of λ. Derive a (1 - α)100% confidence interval for λ. Start with the pivotal statistic and show that for large samples, Z is approximately a standard normal random
Let y̅1 and s12 be the sample mean and sample variance, respectively, of n1 observations randomly selected from a population with mean μ1 and variance σ12. Similarly, define y̅2 and s22 for an independent random sample of n2 observations from a population with mean μ2 and σ22. Derive a
Let (y̅1, s12) and (y̅2 , s22) be the means and variances of two independent random samples of sizes n1 and n2, respectively, selected from normal populations with different means, μ1 and μ2, but with a common variance, σ2. a. Show that E(y̅1 - y̅2) = μ1 - μ2.b. Show thatc. Explain
Refer to Exercise 7.20. According to Theorem 6.11, are independent chi-square random variables with (n1 - 1) and (n2 - 1) df, respectively. Show thatis a chi-square random variable with (n1 + n2 - 2) df.Data from Exercise 7.20Let (y̅1, s12) and (y̅2 , s22) be the means and variances of two
Engineers design tanks for rearing commercial fish to minimize both the use of natural resources (water) and the rearing volume necessary to ensure fish welfare. One key to a well-designed tank is obtaining a reliable estimate of the volume (biomass) of fish reared in the tank. The feasibility of a
A new formula for estimating the water evaporation from occupied swimming pools was proposed and analyzed in the journal, Heating/ Piping/Air Conditioning Engineering (Apr. 2013). The key components of the new formula are number of pool occupants, area of pool’s water surface, and the density
Many ancient Egyptian tombs were cut from limestone rock that contained uranium. Since most tombs are not well ventilated, guards, tour guides, and visitors may be exposed to deadly radon gas. In Radiation Protection Dosimetry (December 2010), a study of radon exposure in tombs in the Valley of
Refer to the Journal of Engineering Education (Jan. 2005) study of the intellectual development of undergraduate engineering students, Exercise 1.27. Intellectual development (Perry) scores were determined for 21 students in a first-year, project-based design course. (Recall that a Perry score of 1
Refer to the Lichen Radionuclide Baseline Research project, Exercise 2.15. University of Alaska researchers determined the amount of the radioactive element cesium-137 for nine lichen specimens sampled from various Alaskan locations. The data values (measured in microcuries per milliliter) are
A group of Harvard University School of Public Health researchers studied the impact of cooking on the size of indoor air particles. (Environmental Science & Technology, September 1, 2000.) The decay rate (measured as mm/hour) for fine particles produced from oven cooking or toasting was
Mechanical engineers have developed a new high-strength aluminum alloy for use in antisubmarine aircraft, tankers, and long-range bombers. (JOM, Jan. 2003.) The new alloy is obtained by applying a retrogression and re aging (RAA) heat treatment to the current strongest aluminum alloy. A series of
Refer to the Harris Corporation/ University of Florida study to determine whether a manufacturing process performed at a remote location can be established locally, Exercise 2.72. Test devices (pilots) were set up at both the old and new locations and voltage readings on 30 production runs at each
Sintering, one of the most important techniques of materials science, is used to convert powdered material into a porous solid body. The following two measures characterize the final product:When Vv = 100%, the product is completely solid—i.e., it contains no pores. Both Vv and Sv are estimated
A team of biomedical engineers designed a prototype device for assisting in obstetric delivery (International Journal for Service Learning in Engineering, Fall 2012). To determine the effectiveness of the device, five medical students were recruited to participate in the study. Initially, each
Research in the International Journal of Foresting Engineering (Vol. 19, 2008) investigated the muscle activity patterns in the neck and upper extremities among forestry vehicle operators. Two types of harvesting vehicles — Timber jack and Valmet — were compared since they differed dramatically
Refer to Exercise 5.45 and the Analytical Chemistry (Dec. 15, 2009) study in which scientists used high-performance liquid chromatography to determine the amount of drug in a tablet. Twenty-five tablets were produced at each of two different, independent sites. Drug concentrations (measured as a
In Kenya, human-induced land-use changes and excessive resource extraction has threatened the jungle ecosystem by reducing animal grazing areas and disrupting access to water sources. In Landscape & Ecology Engineering (Jan. 2013), researchers compared hippopotamus grazing patterns in two
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