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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 Evolutionary Ecology Research (July 2003) study of the New Zealand bird population prior to European contact, Exercise 1.12. Two quantitative variables measured for each of The 116 bird species were body mass (grams) and egg length (millimeters). Descriptive statistics for these
The methodology for conducting a stress analysis of newly designed timber structures is well known. However, few data are available on the actual or allowable stress for repairing damaged structures. Consequently, design engineers often propose a repair scheme (e.g., gluing) without any knowledge
Geneticists at Duke University Medical Center have identified the E2F1 transcription factor as an important component of cell proliferation control (Nature, Sept. 23, 1993). The researchers induced DNA synthesis in two batches of serum-starved cells. Each cell in one batch was micro-injected with
An experiment was conducted to investigate the precision of measurements of a saturated solution of iodine after an extended period of continuous stirring. The data shown in the table represent n = 10 iodine concentration measurements on the same solution. The population variance σ2 measures
The theoretical relationship between heat flux and temperature gradient for homogeneous materials is well known and described by a Fourier equation. However, this relationship does not hold for nonhomogeneous materials such as porous-capillary bodies, cellular systems, suspensions, and pastes. An
Marine biochemists at the University of Tokyo studied the properties of crustacean skeletal muscles (The Journal of Experimental Zoology, Sept. 1993). It is well known that certain muscles contract faster than others. The main purpose of the experiment was to compare the biochemical properties of
Let y̅1 be the mean of a random sample of n1 observations from a Poisson distribution with mean λ1, and let y̅2 be the mean of a random sample of n2 observations from a Poisson distribution with mean λ2. Assume the samples are independent.a. Show that (y̅1 - y̅2) is an unbiased estimator of
Let y1, y2, . . . , yn denote a random sample from a uniform distribution with probability densitya. Show that y̅ is a biased estimator of θ, and compute the bias.b. Find V(y̅).c. What function of y̅ is an unbiased estimator of θ? [1 if e s ys 0 + 1 0 elsewhere
Refer to the IEICE Transactions on Information & Systems (Jan. 2005) comparison of two methods of testing electronic circuits, Exercise 8.50. Recall that each of 11 circuits was tested using the standard compression/depression method and the new Huffman based coding method and the compression
Refer to The International Journal of Pavement Engineering (Sept. 2004) field study of concrete stress at a newly constructed highway, Exercise 8.51. The variable of interest was slab top transverse strain (i.e., change in length per unit length per unit time) at a distance of 1 meter from the
Refer to the AIChE Journal (Jan. 2005) study of a new method for modeling multicomponent transport of gases, Exercise 8.53. Recall that 12 gas mixtures were prepared and the viscosity of each mixture (10-5 Pa.s) was measured both experimentally and with the new model. The results are reproduced in
For the Wilcoxon signed ranks test, show that where n is the number of nonzero differences that are ranked. n(n + 1) T + T = 2
For the special case n = 2, with no ties in the data (that is, no differences of zero), list the eight different ways in which the two absolute differences can be ranked. The number of arrangements, 8, results from the general formula 2n • n!.
For the special case described in Exercise 15.32, show thatFind T+ for each of the eight arrangements of the ranks listed in Exercise 15.31; use the fact that any particular arrangement will occur with probability 1/8. n(n + 1) E(T;) 4
Refer to the Behavioral Neuroscience (Feb. 2004) study of the drug scopolamine’s effects on memory for word-pair associates, Exercise 14.13. Recall that a completely randomized design with three groups was used—group 1 subjects were injected with scopolamine, group 2 subjects were injected with
Refer to the Landscape Ecology Engineering (January 2013) investigation of the impact of soil scouring on the characteristics of overturned and uprooted trees, Exercise 14.12. Recall that a completely randomized design was employed with three treatments (scouring conditions): no scouring (NS),
Food Chemistry (Vol. 106, 2008) published a study of the quality of commercial eggs produced from different housing systems for chickens. Four housing systems were investigated: (1) cage, (2) barn, (3) free range, and (4) organic. Twenty-eight commercial grade A eggs were randomly selected from
Refer to the data on DDT levels of contaminated fish saved in the DDT file. Suppose you want to compare the DDT levels of the three species of fish: (1) channel catfish, (2) smallmouth buffalofish, and (3) largemouth bass.a. Use a graphical method to determine whether a parametric or nonparametric
Refer to the American Journal of Science (Jan. 2005) study of the chemical makeup of buried tills (glacial drifts) in Wisconsin, Exercise 14.11. Recall that till specimens were obtained from five different boreholes (labeled UMRB-1, UMRB-2, UMRB-3, SWRA, and SD), and the ratio of aluminum to
Phosphoric acid is chemically produced by reacting phosphate rock with sulfuric acid. An important consideration in the chemical process is the length of time required for the chemical reaction to reach a specified temperature. The shorter the length of time, the higher the reactivity of the
Use the sum of an arithmetic progression to show that for the Kruskal–Wallis H test,where k is the number of probability distributions being compared and n is the total sample size. п(п + 1) T + T, + .+ T = ...
Refer to the Applied Animal Behavior Science (June 2010) study of stress in cows prior to slaughter, Exercise 14.22. In the experiment, recall that the heart rate (beats per minute) of a cow was measured at four different pre-slaughter phases — (1) first phase of visual contact with pen mates,
The consequences of performing verbal and spatial-imagery tasks on visual search while driving were studied and the results published in the Journal of Experimental Psychology: Applied (Mar. 2000). Twelve drivers were recruited to drive on a highway in Madrid, Spain. During the drive, each subject
Refer to the Journal of Food Engineering (September, 2013) study of the cooling performances fruit container designs, Exercise 14.23. Recall that three container types were investigated— Standard, Supervent, and Ecopack. The containers arranged fruit in either two or three rows; thus, row was
Refer to the Soldering & Surface Mount Technology (Vol. 13, 2001) study to compare four soldering methods, Exercise 14.24. Recall that a measure of plastic hardening (Nm/m2) was obtained for each solder type at each of six different temperatures. The data are reproduced in the table. Analyze
An optical mark reader (OMR) is a machine that is able to “read” pencil marks that have been entered on a special form. A manufacturer of OMRs believes its product can operate equally well in a variety of temperature and humidity environments. To determine whether operating data contradict this
For data sets that contain many missing values, methods for estimating the missing values — called imputation algorithms — may be applied. In the journal, Data & Knowledge Engineering (March 2013), researchers compared several imputation algorithms based on using nearest neighbors to
Refer to the Analytical Chemistry (May 2010) evaluation of a new method of typing blood, Exercise 10.6. Recall that blood drops were applied to the paper and the rate of absorption (called blood wicking) was measured. The next table gives the wicking lengths (millimeters) for six blood drops, each
Refer to the Engineering Structures (July 2013) study of the characteristics of fresh concrete, Exercise 10.38. Recall that the researchers investigated the linear effect of time needed for pressure stabilization (x) on each of three different dependent variables: y1 = initial setting time (hours),
Refer to the Mining Engineering (Oct. 2004) study of a new iron-making technology, Exercise 10.25. Recall that the carbon content produced in a pilot plant test was compared to that from laboratory furnace tests. The data for 25 pilot tests are reproduced in the accompanying table. Conduct a
Refer to Exercise 10.14 and the model of the thermal performance of integral-fin tubes used in the refrigeration and process industries (Journal of Heat Transfer, Aug. 1990). The data in the table (p. 876) are the unflooded area ratio (x) and heat transfer enhancement ( y) values recorded for the
Show that for the special case where n = 3, -1 ≤ rs ≤ 1. List each of the 3! × 3! = 36 different arrangements of the x and y rankings and compute rs for each.
Show that for the special case where n = 3, E (rs) = 0. Use the results of Theoretical Exercise 15.60 and the fact that any of the arrangements has a probability of 1/36 of occurring.
Refer to Exercise 14.81 and the study to compare the speeds of three drill bits. Recall that five drilling sites were randomly assigned to each bit, and the rate of penetration (RoP) in feet per hour was recorded after drilling 3,000 feet at each site. Based on the information given in the table,
The biting rate of a particular species of fly was investigated in a study reported in the Journal of the American Mosquito Control Association (Mar. 1995). Biting rate was defined as the number of flies biting a volunteer during 15 minutes of exposure. This species of fly is known to have a median
Refer to Exercise 15.63. The effect of wind speeds in kilometers per hour (kph) on the biting rate of the flies on Stanbury Island, Utah, was investigated by exposing samples of volunteers to one of six wind speed conditions. The distributions of the biting rates for the six wind speeds were
Refer to Exercise 8.104 and the study to compare human real-time scheduling to an automated approach that utilizes computerized robots and sensing devices (IEEE Transactions, Mar. 1993). Recall that eight simulated scheduling tasks were performed by a human scheduler and by the automated system.
The building specifications in a certain city require that the sewer pipe used in residential areas have a median breaking strength of more than 2,500 pounds per lineal foot. A manufacturer who would like to supply the city with sewer pipe has submitted a bid and provided the following additional
Refer to the EPA study of mercury poisoning in Maine lakes, Exercise 10.77. Lakes can be classified into three trophic states: Oligotrophic lakes have a balance between decaying vegetation and living organisms; eutrophic lakes have a high decay rate in the top layer of water; and, mesotrophic lakes
Refer to Exercise 14.77 and the study to determine the effects of acid rain on the acidity of soils in a natural ecosystem. Recall that experimental plots were irrigated with acid rain at two pH levels: 3.7 and 4.5. The acidity of the soil was then measured at three different depths: 0–15,
A serious, drought-related problem for farmers is the spread of aflatoxin, a highly toxic substance caused by mold, which contaminates field corn. In higher levels of contamination, aflatoxin is potentially hazardous to animal and possibly human health. (Officials of the FDA have set a maximum
Refer to the Journal of Engineering for Industry (Aug. 1993) study of an automated warehouse, Exercise 10.73. Recall that the number of vehicles was varied and the congestion time (total time one vehicle blocked another) was recorded for a simulated automated warehouse. The data are reproduced in
Refer to the Microcomputers in Civil Engineering study of lateral drift in a building, Exercise 14.76. The data shown in the table are the lateral displacements (in inches) estimated by three different computer programs at each of five different building levels. Compare the distributions of lateral
A hybrid algorithm for solving a polynomial zero–one mathematical program was presented in IIE Transactions (June 1990). The algorithm incorporates a mixture of pseudo-Boolean concepts and time-proven implicit enumeration procedures. Twenty five random problems were solved using the hybrid
A preliminary study was conducted to obtain information on the background levels of the toxic substance polychlorinated biphenyl (PCB) in soil samples in the United Kingdom (Chemosphere, Feb. 1986). Such information could then be used as a benchmark against which PCB levels at waste disposal
Synthetic fibers (such as rayon, nylon, and polyester) account for approximately 70% of all fibers used by American mills in their production of textile products. An experiment was conducted to compare the breaking tenacity of synthetic fibers produced using two methods of spinning: wet spinning
Refer to the Corrosion Science (Sept. 1993) study on the resistivity of an amorphous iron– boron–silicon alloy after crystallization, Exercise 10.74. Five alloy specimens were annealed at 700°C, each for a different length of time. The passivation potential—a measure of resistivity of the
Refer to Exercise 10.79 and the straight-line model relating the mean shear strength E(y) of masonry joints to pre compression stress, x. To test this model, a series of stress tests were performed on solid bricks arranged in triplets and joined with mortar (Proceedings of the Institute of Civil
The EPA wants to determine whether temperature changes in the ocean’s water caused by a nuclear power plant will have a significant effect on the animal life in the region. Recently hatched specimens of a certain species of fish are randomly divided into four groups. The groups are placed in
Oil producers are interested in finding high-strength nickel alloys that are corrosion-resistant. Nickel alloys are especially susceptible to hydrogen embrittlement, a process that results when the alloy is cathodically charged in a sulfuric acid solution. To rate the performance of two incoloy
Agent Orange, the code name for a herbicide developed for the U.S. armed forces in the 1960s, was found to be extremely contaminated with TCDD, or dioxin. During the Vietnam War, an estimated 19 million gallons of Agent Orange was used to destroy the dense plant and tree cover of the Asian jungle.
In zoology, the phenomenon of fish moving excessively from one confined area to another is known as excessive transitory migration (ETM). To investigate the ETM of guppy populations, 40 adult female guppies were placed into the left compartment of an experimental aquarium tank that was divided in
Throughout this chapter we have omitted the theoretical derivations of the null distributions of the various nonparametric test statistics. However, we can use computer simulation to derive approximate rejection regions for the tests. Consider the problem of finding the approximate sampling
Follow the steps outlined in Exercise 15.81 to find the approximate critical value (at α = .05) of Spearman’s test for rank correlation for the case n = 10. In part a you will need to randomly order the n = 10 y ranks and n = 10 x ranks.Data from exercise 15.81Throughout this chapter we have
Refer to the Software Quality Professional (Nov. 2004) evaluation of the performance of a software engineering team’s performance at Motorola, Inc., Exercise 5.42. Recall that the variable of interest was the number of updates to a file changed because of a problem report. The monthly number of
Refer to the Mining Engineering (Oct. 2004) study of a new technology for producing high-quality iron nuggets directly from raw iron ore and coal, Exercise 10.25. For one phase of the study, the percentage change in the carbon content of the produced nuggets was measured at 4-hour intervals for 33
Refer to the Analytical Chemistry (Dec. 15, 2009) study of a method—called high performance liquid chromatography—of determining the amount of drug in a tablet, Exercise 5.45. Assume the data in the accompanying table represent the drug concentrations (measured as a percentage) in tablets
In Quality Engineering (Vol. 25, 2013), statisticians at Minitab, Inc., applied control chart methodology to monitor the time between discharges of male patients with hospital-acquired urinary tract infections (URI). The data (days between discharges) for n = 54 consecutive URI discharges at a
Each month, the quality control engineer at a bottle manufacturing company randomly selects one finished bottle from the production process at 20 points in time (days) and records the weight of each bottle (in ounces). The data for last month’s inspection are provided in the next table.a.
Molded-rubber expansion joints, used in heating and air-conditioning systems, are designed to have internal diameters of 5 inches. To monitor the manufacturing process, one joint was randomly selected each hour from the production line and its diameter (in inches) measured, for a period of 12
A rheostat knob, produced by plastic molding, contains a metal insert. The fit of this knob into its assembly is determined by the distance from the back of the knob to the far side of a pin hole. To monitor the molding operation, one knob from each hour’s production was randomly sampled and the
Various interventions are available for nurses to help relieve patients’ pain (e.g., heat/cold applications, breathing exercises, massage). The journal Research in Nursing & Health (Vol. 35, 2012) demonstrated the utility of statistical process control in determining the effectiveness of a
A corporation that manufactures field rifles for the Department of Defense operates a production line that turns out finished firing pins. To monitor the process, an inspector randomly selects five firing pins from the production line, measures their lengths (in inches), and repeats this process at
Refer to the production of molded-rubber expansion joints used in heating and air conditioning systems, Exercise 16.6. To monitor the mean of the manufacturing process, eight joints (rather than one joint) were randomly selected from the production line and their diameters (in inches) measured each
Refer to the manufacture of a rheostat knob, Exercise 16.7. To monitor the process mean, five knobs from each hour’s production were randomly sampled and the distance from the back of the knob to the far side of a pin hole was measured on each. The measurements (in inches) for the first 27 hours
Refer to Exercise 16.8, where the desired circuit path widths were .5 to .85 micron. The manufacturer sampled four CPU chips six times a day (every 90 minutes from 8:00 A.M. until 4:30 P.M.) for five consecutive days. The path widths were measured and used to construct the MINITAB R-chart shown
Refer to the Research in Nursing & Health (Vol. 35, 2012) study of the effectiveness of a pain intervention, Exercise 16.9. Recall that pain levels (measured on a 100-point scale) were recorded for a sample of 10 ICU patients 24-hours post-surgery each week for 20 consecutive weeks. The data
Refer to Exercise 16.11. Suppose the inspector wants to monitor the variation in firing pin lengths with an R-chart.a. Locate the center line for the R-chart.b. Locate upper and lower control limits for the R-chart.c. Calculate and plot the 10 sample ranges in an R-chart. Does the process variation
Construct an R-chart for the data of Exercise 16.12 to monitor the variation in the diameters of the molded-rubber expansion joints produced by the manufacturing process. Does the process appear to be in control?Data from Exercise 16.12Refer to the production of molded-rubber expansion joints used
A soft-drink bottling company is interested in monitoring the amount of cola injected into 16-ounce bottles by a particular filling head. The process is entirely automated and operates 24 hours a day. At 6:00 a.m. and 6:00 p.m. each day, a new dispenser of carbon dioxide capable of producing 20,000
Refer to Exercise 16.16 and the hourly data collected by BPI consulting.a. Construct an R-chart for the data. Is the process variation in control?b. Round each measurement in the data set to a whole number, like in Exercise 16.16b. Form an R-chart for the rounded data. What do you observe?Data from
Examine the sequences of points in parts a–f for any trends.a. + + – – – – – + + + + b. – + – – – + + – + + + +c. – – – – – + + + + + – – d. – +++++ – ++++++++e. + – + + + – – + + – f. – + + + + + + + + + –
Refer to the x̅- and R-charts, Exercises 16.8 and 16.17. Conduct a runs analysis to detect any trend in the process. Data from Exercise 16.8 The central processing unit (CPU) of a microcomputer is a computer chip containing millions of transistors. Connecting the transistors are slender circuit
Refer to the Research in Nursing & Health (Vol. 35, 2012) study of the effectiveness of a pain intervention, Exercises 16.9 and 16.18. Conduct a runs analysis on both the x̅-chart and the R-chart. Interpret the results.Data from Exercise 16.9Various interventions are available for nurses to
Refer to the x̅- and -charts, Exercises 16.11 and 16.20. Conduct a runs analysis to detect any trend in the process.Data from Exercise 16.11A corporation that manufactures field rifles for the Department of Defense operates a production line that turns out finished firing pins. To monitor the
Refer to the x̅- and R-charts, Exercises 16.12 and 16.21. Conduct a runs analysis to detect any trend in the process.Data from Exercise 16.12Refer to the production of molded-rubber expansion joints used in heating and air conditioning systems, Exercise 16.6. To monitor the mean of the
Refer to the x̅- and R-charts, Exercises 16.15 and 16.24. Conduct a runs analysis to detect any trend in the process.Data from Exercise 16.15Refer to the manufacture of a rheostat knob, Exercise 16.7. To monitor the process mean, five knobs from each hour’s production were randomly sampled and
Prestressed concrete cylinder pipe (PCCP) is a rigid pipe designed to take optimum advantage of the tensile strength of steel and the compressive strength and corrosive-inhibiting properties of concrete. PCCP, produced in laying lengths of 24 feet, is susceptible to major stress cracks during the
A manufacturer of computer terminal fuses wants to establish a control chart to monitor the production process. Each hour, for a period of 25 hours, during a time when the process is known to be in control, a quality control engineer randomly selected and tested 100 fuses from the production line.
An electronics company manufactures several types of cathode-ray tubes on a mass production basis. To monitor the process, 50 tubes of a certain type were randomly sampled from the production line and inspected each day over a 1-month period. The number of defectives found each day is provided in
The number of imperfections (scratches, chips, cracks, and blisters) in manufactured custom wood cabinet panels is important both to the customer and to the custom builder. To monitor the manufacturing process, each hour for 15 consecutive hours a finished panel 4 feet by 8 feet was selected and
A quality control study was undertaken by the supervisor of a group of computer programmers. For each of the last 20 days, 25 program modules were randomly selected and inspected for coding errors. The numbers of errors observed per day are recorded in the table.a. Plot the number of coding errors
Refer to Exercise 16.44. The numbers of alignment errors observed for each of the next 25 aircraft produced are shown in the table below.a. Add these 25 points to the c-chart of Exercise 16.44. Does the process still appear to be in control?b. Conduct a runs analysis for the revised c-chart. What
Electronics products (e.g., backplanes, complex motherboards for server systems, etc.) can have as many as thousands of opportunities for defects per printed circuit board (PCB). The defects can be traced to improper solder joints (potentially thousands on a PCB), missing components, improperly
University of Waterloo (Canada) statistician S. H. Steiner applied control chart methodology to the manufacturing of a horseshoe-shaped metal fastener called a robotics clamp (Applied Statistics, Vol. 47, 1998). Users of the clamp were concerned with the width of the gap between the two ends of the
Refer to Exercise 16.11. Use all the sample information to find a 95% tolerance interval for 90% of the firing pin lengths. Assume the distribution of pin lengths is approximately normal.Data from Exercise 16.11A corporation that manufactures field rifles for the Department of Defense operates a
J. Namias used the techniques of statistical quality control to determine when to conduct a search for specific causes of consumer complaints at a beverage company (Journal of Marketing Research, Aug. 1964). Namias discovered that when the process was in control, the biweekly complaint rate of a
Refer to Exercise 16.15. Find a 99% tolerance interval for at least 95% of the distance measurements assuming each of the following:a. A normal distributionb. A non normal distributionData from Exercise 16.15Refer to the manufacture of a rheostat knob, Exercise 16.7. To monitor the process mean,
Many hand tools used by mechanics involve attachments that fit into sockets (e.g., a socket wrench). In the manufacturing of the tools, specifications require that the inside diameter of the socket be larger than the outside diameter of the extension. That is, there must be enough clearance so that
Refer to the Mining Engineering (Oct. 2004) study of a new technology for producing high quality iron nuggets, Exercise 16.2. The data on percent carbon change in produced nuggets for 33 time intervals is saved in the CARBON2 file. Specifications state that the carbon content should be within 3.42
A machine fills boxes with bran flakes. The target weight for the filled boxes is 24 ounces. To monitor the process, five boxes are randomly sampled from each day’s production and weighed. The data for 20 consecutive days is given in the table below. Assume the specification limits for the
A precision parts manufacturer produces bolts for use in military aircraft. The company sampled four consecutively produced bolts each hour on the hour for 25 consecutive hours and measured the length of each bolt. The data on lengths of bolts are shown in the table at the bottom of the page.
Consider a sampling plan with sample size n = 15 and acceptance number α = 1.a. Calculate the probability of lot acceptance for fractions defective p = .1, .2, .3, .4, and .5. Sketch the operating characteristic curve for the plan.b. Find the producer’s risk if AQL = .05.c. Find the consumer’s
Consider a sampling plan with sample size n = 5 and acceptance number a = 0.a. Calculate the probability of lot acceptance for fractions defective p = .1, .3, and .5. Sketch the operating characteristic curve for the plan.b. Find the producer’s risk if AQL = .01.c. Find the consumer’s risk if
The tensile strengths of wires in a certain lot of size 400 are specified to exceed 5 kilograms. Consider an acceptance sampling plan based on a sample of n = 10 wires and acceptance number α = 1.a. Find the producer’s risk if the AQL is 2.5%.b. Find the consumer’s risk if the lot fraction
Refer to Exercise 16.61. Find the appropriate MIL-STD-105D normal (level) general inspection sampling plan for a lot size of 400 wires and an AQL of 2.5%.Data from Exercise 16.61The tensile strengths of wires in a certain lot of size 400 are specified to exceed 5 kilograms. Consider an acceptance
Find the appropriate MIL-STD- 105D general inspection sampling plan for a lot size of 5,000 items and an AQL of 4% under each of the following inspection categories:a. Reduced (I) inspection levelb. Normal (II) inspection levelc. Tightened (III) inspection level
One of the operations in a plant consists of thread-grinding a fitting for an aircraft hydraulic system. To monitor the process, a production supervisor randomly sampled five fittings for each hour, for a period of 20 hours, and measured the pitch diameters of the threads. The measurements,
Suppose the process for manufacturing electrical shafts is in control. At the end of each hour, for a period of 20 hours, the manufacturer randomly selects one shaft and measures the diameter. The measurements (in inches) for the 20 samples are recorded in the table. Construct and interpret a
Specifications require the nickel content of manufactured stainless steel hydraulic valves to be 13% by weight. To monitor the production process, four valves were selected from the production line each hour over an 8-hour period and the percentage nickel content was measured for each, with the
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