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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 Analytical Chemistry (May 2010) study in which medical researchers tested a new method of typing blood using lost cost paper, Exercise 10.6. The data were used to fit the straight-line model relating y = wicking length to x = antibody concentration.a. Give the values of SSE, s2, and s
Calculate SSE, s2, and s for the least-squares lines in:a. Exercise 10.7 b. Exercise 10.8c. Exercise 10.9 d. Exercise 10.10e. Exercise 10.11 f. Exercise 10.12g. Exercise 10.13 h. Exercise 10.14i. Exercise 10.15Interpret the value of s for each line.
The performance of a solar cell can deteriorate when atmospheric dust accumulates on the solar panel surface. In the International Journal of Energy and Environmental Engineering (Dec. 2012), researchers at the Renewable Energy Research Laboratory, University of Lucknow (India) estimated the
Researchers at Dow Chemical Co. investigated the effect of tablet surface area and volume on the rate at which a drug is released in a controlled- release dosage. (Drug Development and Industrial Pharmacy, Vol. 28, 2002.) Six similarly shaped tablets were prepared with different weights and
In a manufacturing process that involves a single machine, decisions must be made on whether to deliver the product to the customer immediately upon completion of the job or to hold the finished product in a batch with other jobs to be delivered at a later time. A computerized mathematical model
Show that V(s2) = 2σ4/(n - 2). The result follows from Theorem 10.1 and the fact that V(x2) = 2v. THEOREM 10.1 Let s = SSE/(n – 2). Then, when the assumptions of Section 10.2 are satisfied, the %3D statistic SSE x = (n – 2)s possesses a chi-square distribution with v = (n – 2) degrees of
Verify that SSE = Σ (yi - ŷi)2 = SSyy - β̂1SSxy.
Refer to the Analytical Chemistry (May 2010) study in which medical researchers tested a new method of typing blood using lost cost paper, Exercises 10.6 and 10.22. Recall that the data was used to fit the straight-line model relating wicking length (millimeters) to x = antibody concentration. A y
Refer to the IHS Journal of Hydraulic Engineering (September 2012) study of water pipes susceptible to breakage, Exercise 10.8. Recall that civil engineers used simple linear regression to model y = the ratio of repair to replacement cost of commercial pipe as a function of x = the diameter (in
Refer to The American Ceramic Society Bulletin (Feb. 2005) evaluation of commercial bricks used in smelter pots, Exercise 10.9. You fit a straight-line model relating the apparent porosity y (in percent) of the brick to mean pore diameter x (in micrometers) using the data provided.a. Find a 95%
Refer to the Chemical Engineering Progress (Jan. 2005) study of the rate at which a spilled volatile liquid will spread across a surface, Exercise 10.10. You fit a straight-line model relating the mass y (in pounds) of the spill to elapsed time x (in minutes) using the data provided.a. Find a 90%
Refer to the Journal of Data Science (Apr. 2004) comparison of methods for estimating rainfall. Exercises 10.11. Consider the simple linear regression relating rain gauge amount (y) to the artificial neural network rain estimate (x).a. Test whether y is positively related to x. Use α = .10.b.
Refer to Exercise 10.12 and the simple linear regression relating the sweetness index (y) of an orange juice sample to the amount of water soluble pectin (x) in the juice. Find a 90% confidence interval for the true slope of the line. Interpret the result.Data from Exercise 10.12The quality of the
Refer to the Journal of Heat Transfer study of the straight-line relationship between heat transfer enhancement, y, and unflooded area ratio, x, Exercise 10.14. Construct a 95% confidence interval for β1, the slope of the line. Interpret the result.Data from Exercise 10.14A study was conducted to
A new method of planning an ecological network was presented in Landscape Ecology Engineering (Jan. 2013). The methodology protects linear green areas that act as ecological corridors for potential movement paths of wild animals (e.g., birds). This requires a prediction of the bird density in the
Engineering Structures (July 2013) published a study of the characteristics of fresh concrete. One key variable studied was the time (in hours) needed for pressure stabilization of the concrete. The researchers investigated the effect of time needed for pressure stabilization (x) on each of three
Ecologists classify the cause of forest fragmentation as either anthropogenic (i.e., due to human development activities such as road construction or logging) or natural in origin (e.g., due to wetlands or wildfire). Conservation Ecology (Dec. 2003) published an article on the causes of
Explain why is normally distributed with mean 0 and variance 1 when the four assumptions of Section 10.2 are satisfied. - Bi BI o/VsS Z =
Use the T statistic in Exercise 10.41 as a pivotal statistic to derive a (1 - α)100% confidence interval for β1.Data from Exercise 10.41It can be shown (proof omitted) that the least-squares estimates, β̂o and β̂1, are independent (in a probabilistic sense) of s2. Use this fact, in
A robust and highly adopted model of human movement is Fitts’ Law. According to Fitts’ Law, the time T required to move and select a target of width W that lies at a distance (or amplitude) A is T = a + b log2(2A/W)where a and b are constants estimated using simple linear regression. The
Refer to The American Ceramic Society Bulletin (Feb. 2005) evaluation of commercial bricks used in smelter pots, Exercise 10.9.a. Find a 95% prediction interval for the apparent porosity percentage y of a brick with a mean pore diameter of x = 10 micrometers. Interpret the result.b. Will a 95%
Refer to the Chemicial Engineering Progress (Jan. 2005) study of the rate at which a spilled volatile liquid will spread across a surface, Exercise 10.10.a. Find a 90% confidence interval for the mean mass, E(y), of all spills with an elapsed time of x = 8 minutes. Interpret the result.b. Will a
Refer to the Journal of Data Science (Apr. 2004) evaluation of methods for estimating rainfall, Exercise 10.11. Find a 99% prediction interval for the rain gauge amount y when the neural network estimate is x = 3 millimeters. Interpret the result.Data from Exercise 10.11Accurate measurements of
Refer to the simple linear regression of sweetness index y and amount of pectin x for n = 24 orange juice samples, Exercise 10.12. The SPSS printout of the analysis is shown below. A 90% confidence interval for the mean sweetness index, E(y), for each value of x is shown below on the SPSS
Refer to the Journal of Heat Transfer (Aug. 1990) study of copper integral fin-tubes, Exercise 10.14. Find a 90% confidence interval for the mean heat transfer coefficient, E(y), when unflooded area ratio is x = 1.95. Interpret the result.Data from Exercise 10.14A study was conducted to model the
In forestry, the diameter of a tree at breast height (which is fairly easy to measure) is used to predict the height of the tree (a difficult measurement to obtain). Silviculturists working in British Columbia’s boreal forest conducted a series of spacing trials to predict the heights of several
To improve the quality of the output of any production process, it is necessary first to understand the capabilities of the process (Deming, Out of the Crisis, 1982). In a particular manufacturing process, the useful life of a cutting tool is linearly related to the speed at which the tool is
Suppose you want to predict some future value of y when x = xp using the prediction equation ŷ = β̂o + β̂1x. The error of prediction will be the difference between the actual value of yp and the predicted value ŷ, i.e.,Error of prediction = yp - ŷa. Explain why the error of
Explain whyis a standard normal random variable. Error or prediction Z = Standard deviation of the error Yp - ý Yp - ý (Xp - X)?
Show thathas a Student’s T distribution with v = (n - 2) df . Then use the T statistic as a pivotal statistic to derive a (1 - α)100% prediction interval for yp. Error of prediction T Estimated standard deviation of the error (xp - 1) 1 + S,
Refer to the Landscape Ecology Engineering (Jan. 2013) study of an ecological network, Exercise 10.37. Recall that the researchers used the data collected for 21 bird habitats to fit the model, E(y) = βo + β1x, where y = bird density and x = vegetation coverage (percentage).a. Use the least
Refer to the American Journal of Physics (June 2010) study of sound waves in a spherical cavity, Exercise 10.54. You fit a straight line model relating frequency (y) of sound waves resulting from striking a basketball with a metal rod to number of resonances (x) and determined the model was
Refer to the study of the relationship between the “sweetness” of orange juice (measured as an index) and the amount of water soluble pectin (parts per million) used in the manufacturing process, Exercise 10.12. You used simple linear regression to predict sweetness index (y) from pectin amount
Refer to the Journal of Heat Transfer (Aug. 1990) study of the relationship between the amount of heat transferred in a copper tube and the area at the top of the tube that is not flooded by condensed vapor, Exercise 10.14. You used simple linear regression to predict the heat transfer enhancement
Refer to the Journal of Civil Engineering and Construction Technology (Feb. 2013) study of bottom ash waste asphalt, Exercise 10.15. Recall that the researchers investigated the relationship between the cracking rate y of bottom ash waste asphalt and stress intensity x on data collected for 15
Refer to the IHS Journal of Hydraulic Engineering (September 2012) study of water pipes susceptible to breakage, Exercise 10.31. Recall that civil engineers used simple linear regression to model y = the ratio of repair to replacement cost of commercial pipe as a function of x = the diameter (in
At temperatures approaching absolute zero (273 degrees below zero Celsius), helium exhibits traits that defy many laws of conventional physics. An experiment has been conducted with helium in solid form at various temperatures near absolute zero. The solid helium is placed in a dilution
High-resolution images of lapetus, one of Saturn’s largest moons, were recently obtained by the Cassini spacecraft and analyzed by NASA (Science, Feb. 25, 2005). Using wideband filters, the ratios of ultraviolet to green and infrared to green wavelengths were measured at 24 moon locations. These
Modern warehouses employ computerized and automated guided vehicles for materials handling. Consequently, the physical layout of the warehouse must be carefully designed to prevent vehicle congestion and optimize response time. Optimal design of an automated warehouse was studied in The Journal of
Amorphous alloys have been found to have superior corrosion resistance. Corrosion Science (Sept. 1993) reported on the resistivity of an amorphous iron–boron–silicon alloy after crystallization. Five alloy specimens were annealed at 700°C, each for a different length of time. The passivation
In response to a health advisory regarding mercury poisoning in Maine lakes, the Environmental Protection Agency conducted a field study of 120 Maine lakes. (Statistical Case Studies: A Collaboration Between Academe and Industry, American Statistical Association, 1998.) In addition to the mercury
The U.S. Department of Agriculture has developed and adopted the Universal Soil Loss Equation (USLE) for predicting water erosion of soils. In geographic areas where runoff from melting snow is common, calculating the USLE requires an accurate estimate of snowmelt runoff erosion. An article in the
Two processes for hydraulic drilling of rock are dry drilling and wet drilling. In a dry hole, compressed air is forced down the drill rods to flush the cuttings and drive the hammer; in a wet hole, water is forced down. An experiment was conducted to determine whether the time y it takes to dry
Civil engineers often use the straight-line equation E(y) = βo + β1x to model the relationship between the mean shear strength E(y) of masonry joints and pre compression stress x. To test this theory, a series of stress tests was performed on solid bricks arranged in triplets and joined with
The Mixed Arithmetic- Perceptual (MA-P) model is a componential model of graphic interaction that was developed based on analyses of humans interacting with graphical displays on the computer. The assumptions of the MA-P model were tested in a research article reported in the SIGCHI Bulletin (July
The Environmental Protection Agency establishes industrial and occupational standards for the ambient air quality of total suspended particulates. The high-volume air sampler—the standard device used for sampling total suspended particulates—collects suspended particulates on large filters. The
Passive and active solar energy systems are becoming viable options to home builders as installation and operating costs decrease. Laminated solar modules utilize high-quality, single-crystal, silicon solar cells, connected electrically in series, to deliver a specified power output. Research was
Refer to The American Ceramic Society Bulletin (Feb. 2005) study of aluminum smelter pots, Exercise 10.9. Recall that the life length of a smelter pot depends on the porosity of the brick lining. The apparent porosity of each of six brick specimens, as well as the mean pore diameter of each brick,
It can be shown (proof omitted) that the least-squares estimates, and , are independent (in a probabilistic sense) of s2. Use this fact, in conjunction with Theorem 10.1 and the result of Exercise 10.40, to show thathas a Student’s T distribution with . B - B. T = s/VS
Refer to the Drug Development and Industrial Pharmacy (Vol. 28, 2002.) investigation of the rate at which a drug is released in a controlled-release dosage, Exercise 10.26. Recall that the ratio of surface area to volume and the diffusional release rate (percentage of drug released divided by the
Refer to the Medical Engineering & Physics (May 2013) study involving the use of fractal geometry to characterize human cortical bone, Exercise 10.13. Experimental data on fractal dimension, x (a measure of the variation in the volume of cortical bone tissue) and Young’s Modulus, y (a measure
An article published in Engineering Structures and Technologies (Sep. 2012) presented the results of a study on the use of processed straw as thermal insulation for homes. Chopped straw specimens were prepared and tested for thermal conductivity at a temperature of 10º Celcius. Two variables were
A hash function is an algorithm that maps data of variable length to data of a fixed length. Hash functions are used by engineers responsible for data authentication and data encryption on a large scale. In the Journal of Cryptographic Engineering (Nov. 2012), a new algorithm for accelerating hash
A study was conducted at Union Carbide to identify the optimal catalyst preparation conditions in the conversion of monoethanolamine (MEA) to ethylenediamine (EDA), a substance used commercially in soaps. For each of 10 preselected catalysts, the following experimental variables were measured:The
”Zoning” is defined as the distribution of vacant land to residential and nonresidential uses via policy set by local governments. Although the negative effects of zoning have been studied (e.g., distorting urban property markets, creating barriers to residential mobility, and impeding economic
Since β̂o, β̂1, ... β̂k are independent of s2, it follows thatl = aoβ̂o + a1β̂1 + ...... + akβ̂kis independent of s2. Use this fact and Theorems 11.2 and 11.3 to show thathas a Student’s T distribution with [n - (k + 1)] degrees of freedom. e - E(€) T = sVa' (X'X)a
Let l = ŷ = β̂o, + β̂1x1 + β̂2x2, ... β̂kxk. Use the T statistic of Exercise 11.13, in conjunction with the pivotal method, to derive the formula for a (1 - α)100% confidence interval for E(y).
Let ŷ = β̂o, + β̂1x1 + β̂2x2, ... β̂kxk be the least squares prediction equation, and let y be some observation to be obtained in the future.a. Explain why (ŷ - y) is normally distributed.b. Show that E(ŷ - y) = 0andV(ŷ - y) = [1 + a'(X'X)-1a]σ2
Show thathas a Student’s T distribution with [n - (k + 1)] degrees of freedom. ŷ - y T = sVI +a'(X'X)-'a
Entanglement of marine mammals (e.g., whales) in fishing gear is considered a significant threat to the species. A study published in Marine Mammal Science (April 2010) investigated the characteristics of whales entangled in fishing nets in the East Sea of Korea. A sample of 207 entanglements of
Scientists believe that a major cause of global warming is higher levels of carbon dioxide (CO2) in the atmosphere. In the Journal of World-Systems Research (Summer 2003), sociologists examined the impact of foreign investment dependence on CO2 emissions n = 66 in developing countries. In
Refer to the Environmental Science & Technology (Jan. 2005) study of the reliability of a commercial kit to test for arsenic in groundwater, Exercise 7.59. Recall that a field kit was used to test a sample of 328 groundwater wells in Bangladesh. In addition to the arsenic level (micrograms per
Refer to the Journal of Engineering for Gas Turbines and Power (Jan. 2005) study of a high-pressure inlet fogging method for a gas turbine engine, Exercise 8.29 (p. 392). Recall that the heat rate (kilojoules per kilowatt per hour) was measured for each in a sample of 67 gas turbines augmented with
Refer to the U.S. Army Corps of Engineers data on fish contaminated from the toxic discharges of a chemical plant located on the banks of the Tennessee River in Alabama. Recall that the engineers measured the length (in centimeters), weight (in grams), and DDT level (in parts per million) for 144
In the oil industry, water that mixes with crude oil during production and transportation must be removed. Chemists have found that the oil can be extracted from the water/oil mix electrically. Researchers at the University of Bergen (Norway) conducted a series of experiments to study the factors
Refer to the U.S. Army Corps of Engineers data on contaminated fish, Exercise 11.26. You fit the first-order model relating DDT level (y) to miles upstream (x1), fish length (x2), and fish weight (x3).a. Propose a model for E(y) that hypothesizes that the rate of increase of DDT level with length
Refer to the Journal of Colloid and Interface Science (Aug. 1995) study of the factors that influence the voltage level (y) required to separate water from oil, Exercise 11.27.a. Consider using only volume (x1) and salinity (x2) to predict y. Write the equation of an interaction model for E(y).b.
Refer to the IHS Journal of Hydraulic Engineering (September, 2012) study of the repair and replacement of water pipes, Exercise 10.8. Recall that a team of civil engineers used regression analysis to model y = the ratio of repair to replacement cost of commercial pipe as a function of x = the
Refer to the IEICE Transactions on Information & Systems (Jan. 2005) experiment to monitor the impedance to leg movement, Exercise 2.46. Recall that engineers attached electrodes to the ankles and knees of volunteers and measured the signal-to-noise ratio (SNR) of impedance changes. For the
Refer to the Landscape Ecology Engineering (Jan. 2013) study of a new method of planning an ecological network, Exercise 10.37. Based on a sample of 21 bird habitats in China, the researchers modeled y = the bird density (number of birds per hectare) as a linear function of x = the percentage of
Refer to the Conservation Ecology (Dec. 2003) study of the causes of fragmentation for 54 South American forests, Exercise 10.39. Recall that ecologists have developed two fragmentation indices for each forest—one index for anthropogenic fragmentation (y) and one for fragmentation from natural
PCBs make up a family of hazardous chemicals that are often dumped, illegally, by industrial plants into the surrounding streams, rivers, or bays. The table below reports the annual concentrations of PCBs (measured in parts per billion) in water samples collected for two consecutive years from 37
Refer to the U.S. Army Corps of Engineers data on fish contaminated from the toxic discharges of a chemical plant located on the banks of the Tennessee River in Alabama. In Exercise 11.26 you fit the first-order model E(y) = βo + β1x1 + β2x2 + β3x3, where y = DDT level in captured fish, x1 =
Refer to the Journal of Engineering for Gas Turbines and Power (Jan. 2005) study of a high-pressure inlet fogging method for a gas turbine engine, Exercise 11.25. Consider, again, the first-order model for heat rate ( y) as a function of speed (x1), inlet temperature (x2), exhaust temperature (x3),
Refer to the Heat Transfer Engineering (Vol. 34, 2013) study of bubble behavior in subcooled flow boiling, Exercises 11.6 and 11.29. You fit an interaction model for bubble density (y) as a function of x1 = mass flux and x2 = heat flux. Conduct a complete residual analysis for the model. Do you
The data in the table below are sales, y, in thousands of dollars per week, for home improvement outlets in each of four cities. The objective is to model sales, y, as a function of traffic flow, adjusting for city-to-city variations that might be due to size or other market conditions. The model
Breakdowns of machines that produce steel cans are very costly. The more breakdowns, the fewer cans produced, and the smaller the company’s profits. To help anticipate profit loss, the owners of a can company would like to find a model that will predict the number of breakdowns on the assembly
The Journal of World-Systems Research (Summer 2003) reported on a study of the link between foreign investments made 16 years earlier and carbon dioxide emissions. The researchers modeled the annual level (y) of CO2 emissions as a function of seven independent variables for n = 66 developing
Periodically, software engineers must provide estimates of their effort in developing new software. In the Journal of Empirical Software Engineering (Vol. 9, 2004), multiple regression was used to predict the accuracy of these effort estimates. The dependent variable, defined as the relative error
A bioengineer wants to model the amount (y) of carbohydrate solubilized during steam processing of peat as a function of temperature (x1), exposure time (x2), and pH value (x3). Data collected for each of 15 peat samples were used to fit the model E(y) =βo + β1x1 + β2x2 + β3x3A
The management of an engineering consultant firm is considering the possibility of setting up its own market research department rather than continuing to use the services of a market research firm. Management wants to know what salary should be paid to a market researcher, based on years of
In a classic research paper, Hamilton (1987) illustrated the multicollinearity problem with an example using the data shown in the next table. The values of x1, x2, and y in the table represent appraised land value, appraised improvements value, and sale price, respectively, of a randomly selected
Refer to the FTC cigarette data of Example 11.14. Recall that the data are saved in the FTC2 file.a. Fit the model E(y) = βo + β1x1 to the data. Is there evidence that tar content (x1) is useful for predicting carbon monoxide content (y)?b. Fit the model E(y) = βo + β2x2 to the data. Is there
Refer to the Journal of Engineering for Industry study of vehicle congestion in an automated warehouse, Exercise 10.73. The data on number of vehicles (x) and congestion time (y) are reproduced in the table. Consider the straight line model E(y) = βo + β1x.a. Construct Y and X matrices for the
Poly (perfluoropropyleneoxide), i.e., PPFPO, is a viscous liquid used extensively in the electronics industry as a lubricant. In a study reported in Applied Spectroscopy (Jan. 1986), the infrared reflectance spectra properties of PPFPO were examined. The optical density (y) for the prominent
Refer to the Applied Spectroscopy study of the optical density (y) for infrared absorption of the liquid PPFPO, Exercise 11.63. In addition to optical density, band frequency (x1) and film thickness (x2) were measured for 12 experiments. (The data are saved in the PPFPO file.)a. Write an
J. Vuorinen carried out a series of experiments to gather information on the coefficient of permeability of concrete (Magazine of Concrete Research, Sept. 1985). In one experiment, the outflow of water from the pores of a concrete specimen after it had been under saturating water pressure for a
Refer to Exercise 11.66. Vuorinen fit the water outflow–time model for each of nine permeability–porosity combinations and used the results to develop a model for the coefficient of permeability of concrete, y. Specifically, he fit the model* E(y) = βo + β1x + β2x2where x1 =
The Journal of Applied Ecology (Vol. 32, 1995) published a study of the feeding habits of baby snow geese. The data on gosling weight change, digestion efficiency, acid-detergent fiber (all measured as percentages) and diet (plants or duck chow) for 42 feeding trials are saved in the SNOWGEESE
Engineers at the University of Massachusetts studied the viability of using semiconductor lasers for solar lighting in spaceborne applications (Journal of Applied Physics, Sept. 1993). A series of n = 8 experiments with quantum-well lasers yielded the following observations on solar pumping
The American Scientist (July–Aug. 1998) published a study of the relationship between self-avoiding and unrooted walks. In a self-avoiding walk you never retrace or cross your own path, whereas an unrooted walk is a path in which the starting and ending points are impossible to distinguish. The
Chemical engineers at Tokyo Metropolitan University analyzed urban air specimens for the presence of low molecular weight dicarboxylic acid (Environmental Science & Engineering, Oct. 1993). The dicarboxylic acid (as a percent of total carbon) and oxidant concentrations for 19 air specimens
Moissanite is a popular abrasive material because of its extreme hardness. Another important property of moissanite is elasticity. The elastic properties of the material were investigated in the Journal of Applied Physics (Sept. 1993). A diamond anvil cell was used to compress a mixture of
Phosphorus used in soil fertilizers can contaminate freshwater sources during rainfall runoff. Consequently, it is important for water-quality engineers to estimate the amount of dissolved phosphorus in the water. Geoderma (June 1995) presented an investigation of the relationship between soil loss
Refer to the surface albedo study of pond ice, Exercise 11.43. Recall that you fit a second-order model for broadband surface albedo level (y) as a function of pond depth (x) to data saved in the PONDICE file. Conduct a complete residual analysis of the model. Do you recommend any model
Sintering, one of the most important techniques of materials science, is used to convert a 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
Researchers at the University of Aberdeen (Scotland) developed a statistical model for estimating the chemical composition of water (Journal of Agricultural, Biological, and Environmental Statistics, March 2005). For one application, the nitrate concentration (milligrams per liter) in a water
The performance of a diesel engine with blends of biodiesel fuels was the topic of research in the International Journal of Energy and Environmental Engineering (Dec. 2013). Several of the many variables measured in the study are listed below. Identify the type (quantitative or qualitative) of each
The behavior and design of cold-formed steel buildings and walls was investigated in the Journal of Structural Engineering (May 2013). Several of the many variables measured in the study are listed below. Identify the type (quantitative or qualitative) of each variable. (Note: The dependent
In the International Journal of Engineering Research & Applications (May-June 2013), structural engineers examined the properties of Portland cement mortar made from rice ash husk. The variables measured in the study included the following. Identify the type (quantitative or qualitative) of

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