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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
A craps player who has experienced a long run of bad luck at the craps table wants to test whether the casino dice are “loaded,” i.e., whether the proportion of “sevens” occurring in many tosses of the two dice is different from (if the dice are fair, the probability of tossing a
Pesticides applied to an extensively grown crop can result in inadvertent areawide air contamination. Environmental Science & Technology (Oct. 1993) reported on air deposition residues of the insecticide diazinon used on dormant orchards in the San Joaquin Valley, California. Ambient air
A large steel corporation conducted an experiment to compare the average iron contents of two consignments of lumpy iron ore. In accordance with industrial standards, n increments of iron ore were randomly selected from each consignment and measured for iron content. From previous experiments, it
The Journal of Environmental Engineering (Feb. 1986) reported on a heat transfer model designed to predict winter heat loss in wastewater treatment clarifiers. The analysis involved a comparison of clear-sky solar irradiation for horizontal surfaces at different sites in the Midwest. The day-long
Jitter is a term used to describe the variation in conduction time of a modular pulsed-water power system. Low throughput jitter is critical to successful waterline technology. An investigation of throughput jitter in the plasma opening switch of a prototype system (Journal of Applied Physics,
By law, all new cars must be equipped with both driver-side and passenger side safety air bags. There is concern, however, over whether air bags pose a danger for children sitting on the Passenger side. In a National Highway Traffic Safety Administration (NHTSA) study of 55 people killed by the
According to a study conducted by the California Division of Labor Research and Statistics, roofing is one of the most hazardous occupations. Of 2,514 worker injuries that caused absences for a full workday or shift after the injury, 23% were attributable to falls from high elevations on level
In Ireland, the majority of commercial forests are located in remote areas on predominantly peat soils. These roads exhibit rapid deterioration when traversed by logging vehicles or other heavy machinery. A study of the attributes of forest access roads in Ireland was published in the International
Chemical engineers at the University of Murcia (Spain) conducted a series of experiments to determine the most effective membrane to use in a passive sampler (Environmental Science & Technology, Vol. 27, 1993). The effectiveness of a passive sampler was measured by the sampling rate, recorded
Let y1, y2, y3, . . . , yn, represent a random sample of size n selected from a normal probability distribution with unknown mean μ and variance σ2 = 1. Then the sample mean, y̅ , has a normal distribution with mean μ and variance σ2 = 1/n. Assume that the prior distribution for μ is a normal
Let y1, y2, y3, . . . , yn, represent a random sample of size n selected from a Poisson probability distribution with unknown mean λ. Let X represent the sum of the Poisson values, X = ∑Yi . Then X has a Poisson distribution with mean nλ. Assume that the prior distribution for λ
In fields that require high precision surface height maps, white light interferometry (WLI) has become the standard inspection tool. Because WLI generates two-dimensional height profiles, and standard mechanical devices used with WLI generate only one-dimensional profiles, engineers must estimate
Refer to the American Journal of Science (Jan. 2005) study of the chemical makeup of buried glacial drifts (or tills) in Wisconsin, Exercise 7.64. The data on the ratios of aluminum (Al) to beryllium (Be) in sediment for a sample of 26 buried till specimens is reproduced in the table.a. Use the
Refer to the Environmental Science & Technology (Jan. 2005) study of contaminated well sites located near a New Jersey gasoline service station, Exercise 7.29. The data (parts per billion) on the level of methyl tert-butyl ether (MTBE) at 12 sampled sites are reproduced in the table. Use the
Refer to the Composites Fabrication Magazine (Sept. 2004) study of the strength of fiber-reinforced polymer (FRP) composite materials, Exercise 2.47. Recall that 10 specimens of pultruded FRP strips were mechanically fastened to highway bridges and tested for bearing strength. The strength
When determining the sample size required to estimate p, show that the sample size n is largest when p = .5.
Refer to the JOM (Jan. 2003) comparison of a new high-strength RAA aluminum alloy to the current strongest aluminum alloy, Exercise 7.43. Suppose the researchers want to estimate the difference between the mean yield strengths of the two alloys to within 15 MPa using a 95% confidence interval. How
Refer to the Environmental Science & Technology (Jan. 2005) study of methyl tert-butyl ether (MTBE) contamination in New Hampshire wells, Exercise 7.66. How many public and how many private wells must be sampled in order to estimate the difference between the proportions of wells with a
Refer to Exercise 7.35. Suppose that we want to estimate the average decay rate of fine particles produced from oven cooking or toasting to within .04 with 95% confidence. How large a sample should be selected?Data from Exercise 7.35A group of Harvard University School of Public Health researchers
Refer to the Journal of Chemical Ecology (March 2013) study of the effectiveness of pheromones to attract two different strains of fall armyworms, Exercise 7.68. Recall that both corn-strain and rice-strain male armyworms were released into a corn field containing the pheromone and the percentage
Refer to the Computers & Security (July 2013) study of security issues with Microsoft products, Exercise 7.58. Determine the number of security bulletins to be sampled in order for Microsoft to estimate the proportion of bulletins that report a problem with Windows to within .075 with 90%
Refer to The International Journal of Robotics Research (Dec. 2004) algorithm for estimating the sensor motion of a robotic arm, Exercise 2.64. A key variable is the error of estimating arm rotation (measured in radians). Data on rotation error for 11 experiments with different combinations of
Refer to the Journal of Food Engineering (Sep. 2013) study of the characteristics of fried sweet potato chips, Exercise 7.78. Recall that a sample of 6 sweet potato slices fried at 130º using a vacuum fryer yielded the following statistics on internal oil content (measured in gigagrams): y̅1 =
Refer to the Landscape & Ecology Engineering (Jan. 2013) study of hippopotamus grazing patterns in Kenya, Exercise 7.41. Recall that plots of land were sampled in two areas — a national reserve and a pastoral ranch — and the number of hippo trails from a water source was determined for each
Refer to Exercise 7.39 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. Recall that 25 tablets were produced at each of two different, independent sites. The researchers want to determine
Refer to the Environmental Geology (Vol. 58, 2009) simulation study of how far a block from a collapsing rock wall will bounce down a soil slope, Exercise 2.29. Rebound lengths (in meters) were estimated for 13 rock bounces. The data are repeated in the table below. A MINITAB analysis of the data
The Occupational Safety & Health Administration requires companies that handle hazardous chemicals to complete material safety data sheets (MSDS). These MSDS have been criticized for being too hard to understand and complete by workers. A study of 150 MSDS revealed that only 11% were
Use the pivotal statistic T given in Exercise 7.22 to derive a (1 - α)100% small-sample confidence interval for (μ1 - μ2).Data from Exercise 7.22Refer to Exercises 7.20 and 7.21. The pooled estimator of the common variance σ2 is given byData from Exercise 7.20Let (y̅1, s12) and (y̅2 , s22) be
Refer to Exercises 7.20 and 7.21. The pooled estimator of the common variance σ2 is given byShow thathas a Student’s T distribution with (n1 + n2 - 2) df. Recall that T = Z/√x2/v has a Student’s T distribution with v df and use the results of Exercises 7.20c and 7.21.Data from Exercise
Let y1, y2, . . . , yn be a random sample of n observations from an exponential distribution with mean β. Derive a large-sample confidence interval for β. Start with the pivotal statisticand show that for large samples, Z is approximately a standard normal random variable. Then substitute y̅ for
Suppose the random variable Y has a binomial distribution with parameters n and p.a. Show that p̂ = y/n is an unbiased estimator of p.b. Find the variance of p̂.
Let y1, y2, y3, . . . , yn be a random sample from a Poisson distribution with mean λ, i.e., E (yi) = λ, i = 1, 2, .... n. Consider four estimators of λ:a. Which of the four estimators are unbiased?b. Of the unbiased estimators, which has the smallest variance? Recall that, for a Poisson
Let y1, y2, y3 be a random sample from an exponential distribution with mean θ, i.e., E (yi) = θ, i = 1, 2, 3. Consider three estimators of θ:a. Show that all three estimators are unbiased. b. Which of the estimators has the smallest variance? Recall that, for an exponential distribution,
The continuous random variable Y is said to have a lognormal distribution with parameters μ and σ if its probability density function, f(y), satisfiesShow that X = ln (Y) has a normal distribution with mean μ and variance σ2. 1 (In y — м)* f(y) exp oyV27 (y > 0) 202
Use Theorem 6.7 to draw a random sample of n = 5 observations from a population with probability density function given by 2ye- if 0 < y < f(y) : elsewhere
Use Theorem 6.7 to draw a random sample of n = 5 observations from a population with probability density function given by S2(y – 1) if 1 sy < 2 f(y) lo elsewhere
Let Y1 and Y2 be a random sample of n = 2 observations from a normal distribution with mean μ and variance σ2. a. Show thathas a standard normal distribution. b. Given the result in part a, show that Z2 possesses a χ2 distribution with 1 degree of freedom. First show that S2 = (Y1 -
The lifetime Y of an electronic component of a laptop computer has a Rayleigh density, given byFind the probability density function for W = Y2, and identify the type of density function. 2y -3/B if y > 0 f(y) elsewhere
Let X and Y be two continuous random variables with joint density f(x, y). Show that f2(y Ι x)f1(x) = f1(x Ι y)f2(y)
The waiting time Y until delivery of a new component for a data-processing unit is uniformly distributed over the interval from 1 to 5 days. The cost C (in hundreds of dollars) of this delay to the purchaser is given by C = (2Y2 + 3). Find the probability that the cost of delay is at least $2,000,
Refer to the problem of transporting neutral particles in a nuclear fusion reactor, described in Exercise 3.36. Recall that particles released into a certain type of evacuated duct collide with the inner duct wall and are either scattered (reflected) with probability .16 or absorbed with
Let Y̅1 and Y̅2 be the means of independent random samples of sizes n1 and n2 selected from normally distributed populations with parameters (μ1, σ2) and (μ2, σ2), respectively. Ifshow thathas a Student’s T distribution with v = (n1 + n2 - 2) degrees of freedom. (n - 1)S} + (n, - 1)S
University of Maryland engineers investigated the life lengths of several commercial avionics, including flight control systems, autopilots, flight director systems, and symbol generators. (The Journal of the Reliability Analysis Center, 1st Quarter, 2005.) The life length of integrated circuits
Use the results of Exercise 4.39 and the fact thatto find E(Y2) for a binomial random variable. ELY(Y – 1)] - E(Y² – Y) = E(Y²) – E(Y) = E(Y?) – H
Prove Theorem 4.3.
Prove Theorem 4.2. The proof follows directly from Definition 4.5.
Prove Theorem 4.1. Use the fact that ∑all P(y) = 1
Digital signal and image processing (DSIP) has a wide variety of applications, including entertainment (video on demand), telemedicine, security/surveillance, military target recognition, wireless communications, and intelligent transportation systems. Consequently, there is a rapidly growing need
Refer to the Human Factors (Dec. 1988) study of color brightness as a body orientation clue, Exercise 7.65. Ninety college students, reclining on their backs in the dark, were disoriented when positioned on a rotating platform under a slowly rotating disk that blocked their field of vision. The
The American Association of Engineering Societies (AAES) hired Harris Interactive to conduct a survey of the American public’s knowledge of and interest in engineering. (AAES/Harris Poll “American Perspectives on Engineers and Engineering: Final Report.” 13 Feb. 2004.) The primary objective
Because of erratic rainfall patterns and low water-holding capacities of soils in Florida, supplemental irrigation is required for producing most crops. A research team has developed five alternative water-management strategies for irrigating cropland in central Florida. A random sample of 100
For the multinomial probability distribution, show thatCov (ni, nj) = -npipjFirst show that E (ninj) = n(n - 1)pipj.
Refer to the American Association of Engineering Societies (AAES) survey of the public’s knowledge of engineering, Exercise 9.7. Responses to the statement, “Engineers are responsible for creating things that are harmful to society” are summarized and reproduced in the table. Use these
A group of environmental engineers are studying mushroom-like slime molds as a potential food source for insects. (Journal of Natural History, May 2010.) In particular, they investigated which of six species of slime molds are most attractive to beetles inhabiting an Atlantic rain forest. A sample
Refer to the Human Factors (Dec. 1988) study of orientation clues, Exercise 9.6. Conduct a test to compare the proportions of subjects that fall in the three disk-orientation categories. Assume you want to determine whether the three proportions differ. Use α = .05.Data from Exercise 9.6Refer to
Geneticists at Australian National University are studying whether the cognitive effects of Alzheimer’s disease can be detected at an early age (Neuropsychology, Jan. 2007.) One portion of the study focused on a particular strand of DNA extracted from each in a sample of 2,097 young adults
A contract engineer at DuPont Corp. studied the rate at which a spilled volatile liquid will spread across a surface (Chemical Engineering Progress, Jan. 2005). Assume 50 gallons of methanol spills onto a level surface outdoors. The engineer used derived empirical formulas (assuming a state of
A study was conducted to model the thermal performance of integral-fin tubes used in the refrigeration and process industries (Journal of Heat Transfer, Aug. 1990). Twenty-four specially manufactured integral-fin tubes having rectangular shaped fins made of copper were used in the experiment. Vapor
Show that Note thatsince Σ(xi - x̅) = 0. T(x - I) Bo = y - Bx = 2
An innovative new ironmaking technology (called ITmk3) produces high-quality iron nuggets directly from raw iron ore and coal. Mining Engineering (Oct. 2004) published the results of pilot tests conducted on the new process. For one phase of the study, the carbon content produced in a pilot plant
Refer to the Wind Engineering (Jan. 2004) study of two types of timber radiata pine and hoop pine—used in high-efficiency small wind turbine blades, Exercise 10.16. Data on stress (y) and the natural logarithm of number of blade cycles (x) for each timber type were analyzed using simple linear
The standard method of determining the water content of soil involves the removal of the soil and estimation of soil volume. Forest Engineering (July 1999) presented a method that does not require soil removal, i.e., an indirect method, called the radiation method. The new method utilizes the fact
Use the result of Exercise 10.51 to show that rVn - 2 s/VSSx
A general proof of the fact that χ 2 possesses approximately a chi-square sampling distribution when n is large is beyond the scope of this text. However, it can be justified for the binomial case (k = 2). In Optional Exercise 6.118 (p. 259), we stated that if Z is a standard normal random
In Tectonics (Oct. 2004), geologists published their research on the formation of oroclines (curved mountain belts) in the central Appalachian mountains. A comparison was made of two nappes (sheets of rock that have moved over a large horizontal distance), one in Pennsylvania and the other in
Refer to the Environmental Science & Technology (Jan. 2005) study of methyl tert-butyl ether (MTBE) contamination in public and private New Hampshire wells, Exercise 2.12. Recall that data on well class (public or private), aquifer (bedrock or unconsolidated), and detectable level of MTBE
Refer to the AAES survey of the American public’s knowledge of and interest in engineering, Exercise 9.7. Another survey question asked, “What media sources do you use to follow stories about engineering and engineers?” Responses were categorized as either “Follow on the internet” or
The PROMISE Software Engineering Repository, hosted by the School of Information Technology and Engineering, University of Ottawa, provides researchers with data sets for building predictive software models. Data on 498 modules of software code written in C language for a NASA spacecraft instrument
According to the Highway Loss Data Institute (HLDI), “hybrid [automobiles] have a safety edge over their conventional twins when it comes to shielding their occupants from injuries in crashes”. (HLDI Bulletin, Sept. 2011.) Consider data collected by the HLDI on Honda Accords over the past eight
Westinghouse Electric Company has experimented with different means of evaluating the performance of solder joint inspectors. One approach involves comparing an individual inspector’s classifications with those of the group of experts that comprise Westinghouse’s Work Standards Committee. In
When bonding teeth, orthodontists must maintain a dry field. A new bonding adhesive (called Smart bond) has been developed to eliminate the necessity of a dry field. However, there is concern that the new bonding adhesive may not stick to the tooth as well as the current standard, a composite
Seldane-D, produced by Marion Merrell Dow, Inc., is an over-the-counter drug designed to relieve sneezing, nasal congestion, and other symptoms of allergic rhinitis. General adverse effects of Seldane-D were investigated in a double-blind, controlled study of over 500 patients suffering from
Refer to the Disasters (Vol. 28, 2004) study of the effects of a tropical cyclone on the quality of drinking water on a remote Pacific island, Exercise 1.11. One part of the study evaluated the usefulness of a simple paper-strip, hydrogen sulphide (H2S) test kit for water quality in determining the
Refer to the Trends in Biomaterials & Artificial Organs (Jan. 2003) study of a new bonding adhesive for teeth, Exercise 9.33. Recall that the new adhesive (called Smart bond) was compared to the standard composite adhesive. The Adhesive Remnant Index (ARI) scores for 10 teeth bonded with the
Refer to the study on predicting defects in software code written in C language for a NASA spacecraft instrument, Exercise 9.29. The SPSS contingency table for the two categorical variables, actual defective status and predicted defective status using EVG, is reproduced.a. Show that there are 11
Refer to the NAWIC Journal of Professional Issues in Engineering Education & Practice (April 2013) study of the job satisfaction of members of the National Association of Women in Construction (NAWIC), Exercise 9.24. The results for the survey of 477 women are reproduced in the accompanying
A traffic study found that of 972 automobiles entering a busy intersection during the period from 4 P.M. to 7 P.M., 357 turned left, 321 turned right, and 294 drove straight through the intersection. a. Construct a one-way table for the study. b. Find a 95% confidence interval for the
Compressed work weeks are defined as “alternative work schedules in which a trade is made between the number of hours worked per day, and the number of days worked per week, in order to work the standard number of weekly hours in less than 5 days.” A field study was conducted at a large,
Researchers at the Oak Ridge (Tennessee) National Laboratory have developed an algorithm to estimate the numbers of expected and excess cases of thyroid cancer occurring in the lifetime of those exposed to atomic weapons tests at the Nevada Test Site in the 1950s (Health Physics, Jan. 1986). Of the
Four pesticides used in dormant California orchards are chlorpyrifos, diazinon, methidathion, and parathion. Environmental Science & Technology (Oct. 1993) reported the number of applications of these spray chemicals over a 6-month period in California. The data for each of three types of fruit
Refer to the Nature (Sept. 1993) study of animal and plant species “hot spots” in Great Britain, Exercise 3.81. A hot spot is defined as a 10-km square area that is species-rich, i.e., that is heavily populated by the species of interest. Similarly, a cold spot is a 10-km square area that is
Refer to the survey of agricultural engineers, Exercise 9.8. Do the data present sufficient evidence to indicate a preference for one or more of the five water-management strategies? Test using α = .05.Data from Exercise 9.8Because of erratic rainfall patterns and low water-holding capacities of
Dear enemy recognition (DER) is the term used by naturalists and ecologists for the aggressive behavior of birds, mammals, and ants when their territorial boundaries are violated by one of their own species. DER is often followed by escalated attacks on the invading animal. A study explored the
Video engineers use time compression to shorten the time required for broadcasting a television commercial. But can shorter commercials be effective? To answer this question, 200 college students were randomly divided into three groups. The first group (57 students) was shown a videotape of a
A decision support system (DSS) is a computerized system designed to aid in the management and analysis of large data sets. Ideally, a DSS should include four components: (1) a data extraction system, (2) a relational database organization, (3) analysis models, and (4) a user-friendly interactive
An experiment was conducted to compare the fidelity and selectivity of radio receivers. Thirty receivers were tested and classified as low or high in each of the two categories. Do the data in the table provide sufficient evidence to indicate a dependence between fidelity and selectivity? Use
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
Does the propensity for worker injuries depend on the length of time that a worker has been on the job? An analysis of 714 worker injuries by one manufacturer gave the results shown in the table for the distribution of injuries over the eight 1-hour time periods per shift.a. Do the data imply that
The scarce and essential metal, manganese, has been found in abundance in nodules on the deep seafloor. To investigate the relationship between the magnetic age of Earth’s crust on the ocean floor and the probability of finding manganese nodules in that location, crust specimens were selected
The equation for a straight line (deterministic) is y = β̂o + β̂1xIf the line passes through the point (0, 1), then x = 0, y = 1must satisfy the equation. That is, 1 = βo + β1(0)Similarly, if the line passes through the point (2, 3), then x = 2, y = 3 must satisfy the equation:3 =
In each case find the equation of the line that passes through the points. Graph each line.a. (0, 2) and (2, 6)b. (0, 4) and (2, 6)c. (0, -2) and (-1, -6)d. (0, -4) and (3, -7)
Plot the following lines; give the y-intercept and slope of each. a. y = 3 + 2xb. y = 1 + xc. y = - 2 + 3xd. y = 5xe. y = 4 - 2x
In Analytical Chemistry (May 2010), chemical engineers tested a new method of typing blood using lost cost paper. Blood drops were applied to the paper and the rate of absorption (called blood wicking) was measured. The table below gives the wicking lengths (millimeters) for six blood drops, each
Pipes used in a water distribution network are susceptible to breakage due to a variety of factors. When pipes break, engineers must decide whether to repair or replace the broken pipe. A team of civil engineers used regression analysis to estimate y = the ratio of repair to replacement cost of
Accurate measurements of rainfall are critical for many hydrological and meteorological projects. Two standard methods of monitoring rainfall use rain gauges and weather radar. Both, however, can be contaminated by human and environmental interference. In the Journal of Data Science (Apr. 2004).
A common byproduct of burning municipal solid waste is bottom ash. The use of bottom ash waste in the building of asphalt roads was investigated in the Journal of Civil Engineering and Construction Technology (Feb. 2013). Of particular interest was the relationship between the cracking rate of
The maximum likelihood estimator of the mean m of a normal distribution is the sample mean . Consider the model E(y) = μ. Show that the least-squares estimator of μ is also y̅. Minimize SSE = Σ(yi - µ̂ )2 with respect to µ̂.
Consider the pair of simultaneous linear equations:Derive the formulas for the least-squares estimates, β̂o and β̂1. nBo + BiE = Ey Bo Ex + BI E = Exyi
We showed in Example 10.2 that β̂1, the least-squares estimator of the slope b1, is an unbiased estimator of β1, i.e., E(β̂1) = β1. Use the result from Exercise 10.17 to show that E(β̂o) = βo.
In Exercise 10.19, you showed that β̂o could be written as a linear function of independent random variables. Use Theorem 6.8 to show that THEOREM 6.8 The Expected Value E(C) and variance V(O* of a Linear Function of Y, Y2, .. Yn Suppose the means and variances of Y1, Y2. ... Y4 are (u1, oi).
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