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applied statistics and probability for engineers

Probability And Statistics For Engineering And The Sciences 7th Edition Dave Ellis, Jay L Devore - Solutions

Let X1, X2, . . . , Xn be a random sample from a continuous probability distribution having median ~(so that P(Xi ~) P(Xi ~) .5).a. Show that P(min(Xi) ~ max(Xi)) 1 1 2n1 so that (min(xi), max(xi)) is a 100(1 )% confidence interval for ~with 1 2n1. [Hint: The
In Example 6.8, we introduced the concept of a censored experiment in which n components are put on test and the experiment terminates as soon as r of the components have failed. Suppose component lifetimes are independent, each having an exponential distribution with parameter . Let Y1 denote the
Chronic exposure to asbestos fiber is a well-known health hazard. The article “The Acute Effects of Chrysotile Asbestos Exposure on Lung Function” (Environ. Research, 1978: 360–372) reports results of a study based on a sample of construction workers who had been exposed to asbestos over a
A manufacturer of college textbooks is interested in estimating the strength of the bindings produced by a particular binding machine. Strength can be measured by recording the force required to pull the pages from the binding. If this force is measured in pounds, how many books should be tested to
It is important that face masks used by firefighters be able to withstand high temperatures because firefighters commonly work in temperatures of 200–500°F. In a test of one type of mask, 11 of 55 masks had lenses pop out at 250°. Construct a 90% CI for the true proportion of masks of this type
Aphid infestation of fruit trees can be controlled either by spraying with pesticide or by inundation with ladybugs. In a particular area, four different groves of fruit trees are selected for experimentation. The first three groves are sprayed with pesticides 1, 2, and 3, respectively, and the
High concentration of the toxic element arsenic is all too common in groundwater. The article “Evaluation of Treatment Systems for the Removal of Arsenic from Groundwater”(Practice Periodical of Hazardous, Toxic, and Radioactive Waste Mgmt., 2005: 152–157) reported that for a sample of n 5
The financial manager of a large department store chain selected a random sample of 200 of its credit card customers and found that 136 had incurred an interest charge during the previous year because of an unpaid balance.a. Compute a 90% CI for the true proportion of credit card customers who
A journal article reports that a sample of size 5 was used as a basis for calculating a 95% CI for the true average natural frequency (Hz) of delaminated beams of a certain type. The resulting interval was (229.764, 233.504). You decide that a confidence level of 99% is more appropriate than the
For each of 18 preserved cores from oil-wet carbonate reservoirs, the amount of residual gas saturation after a solvent injection was measured at water flood-out. Observations, in percentage of pore volume, were 23.5 31.5 34.0 46.7 45.6 32.5 41.4 37.2 42.5 46.9 51.5 36.4 44.5 35.7 33.5 39.3 22.0
A triathlon consisting of swimming, cycling, and running is one of the more strenuous amateur sporting events. The article “Cardiovascular and Thermal Response of Triathlon Performance” (Medicine and Science in Sports and Exercise, 1988: 385–389) reports on a research study involving nine
Example 1.10 introduced the accompanying observations on bond strength.11.5 12.1 9.9 9.3 7.8 6.2 6.6 7.0 13.4 17.1 9.3 5.6 5.7 5.4 5.2 5.1 4.9 10.7 15.2 8.5 4.2 4.0 3.9 3.8 3.6 3.4 20.6 25.5 13.8 12.6 13.1 8.9 8.2 10.7 14.2 7.6 5.2 5.5 5.1 5.0 5.2 4.8 4.1 3.8 3.7 3.6 3.6 3.6a. Estimate true average
The results of a Wagner turbidity test performed on 15 samples of standard Ottawa testing sand were (in microamperes)26.7 25.8 24.0 24.9 26.4 25.9 24.4 21.7 24.1 25.9 27.3 26.9 27.3 24.8 23.6a. Is it plausible that this sample was selected from a normal population distribution?b. Calculate an upper
The following observations were made on fracture toughness of a base plate of 18% nickel maraging steel [“Fracture Testing of Weldments,” ASTM Special Publ. No. 381, 1965:328–356 (in ksi in., given in increasing order)]:69.5 71.9 72.6 73.1 73.3 73.5 75.5 75.7 75.8 76.1 76.2 76.2 77.0 77.9
Determine the following:a. The 95th percentile of the chi-squared distribution with 10b. The 5th percentile of the chi-squared distribution with 10c. P(10.98 2 36.78), where 2 is a chi-squared rv with 22d. P(2 14.611 or 2 37.652), where 2 is a chisquared rv with 25
Determine the values of the following quantities:a. 2.1,15b. 2.1,25c. 2.01,25d. 2.005,25e. 2.99,25f. 2.995,25
A sample of 25 pieces of laminate used in the manufacture of circuit boards was selected and the amount of warpage(in.) under particular conditions was determined for each piece, resulting in a sample mean warpage of .0635 and a sample standard deviation of .0065.a. Calculate a prediction for the
A study of the ability of individuals to walk in a straight line(“Can We Really Walk Straight?” Amer. J. of Physical Anthro., 1992: 19–27) reported the accompanying data on cadence (strides per second) for a sample of n 20 randomly selected healthy men..95 .85 .92 .95 .93 .86 1.00 .92 .85
The n 26 observations on escape time given in Exercise 36 of Chapter 1 give a sample mean and sample standard deviation of 370.69 and 24.36, respectively.a. Calculate an upper confidence bound for population mean escape time using a confidence level of 95%.b. Calculate an upper prediction bound
Silicone implant augmentation rhinoplasty is used to correct congenital nose deformities. The success of the procedure depends on various biomechanical properties of the human nasal periosteum and fascia. The article “Biomechanics in Augmentation Rhinoplasty” (J. of Med. Engr. and Tech., 2005:
The article “Measuring and Understanding the Aging of Kraft Insulating Paper in Power Transformers” (IEEE Electrical Insul. Mag., 1996: 28–34) contained the following observations on degree of polymerization for paper specimens for which viscosity times concentration fell in a certain middle
A random sample of n 8 E-glass fiber test specimens of a certain type yielded a sample mean interfacial shear yield stress of 30.2 and a sample standard deviation of 3.1 (“On Interfacial Failure in Notched Unidirectional Glass/Epoxy Composites,” J. of Composite Materials, 1985:
Determine the t critical value for a lower or an upper confidence bound for each of the situations described in Exercise 30.
Determine the t critical value that will capture the desired t curve area in each of the following cases:a. Central area .95, df 10b. Central area .95, df 20c. Central area .99, df 20d. Central area .99, df 50e. Upper-tail area .01, df 25f. Lower-tail area .025, df 5
Determine the values of the following quantities:a. t.1,15b. t.05,15c. t.05,25d. t.05,40e. t.005,40
Reconsider the CI (7.10) for p, and focus on a confidence level of 95%. Show that the confidence limits agree quite well with those of the traditional interval (7.11) once two successes and two failures have been appended to the sample [i.e., (7.11) based on x 2 S’s in n 4 trials].[Hint: 1.96
The superintendent of a large school district, having once had a course in probability and statistics, believes that the number of teachers absent on any given day has a Poisson distribution with parameter . Use the accompanying data on absences for 50 days to derive a large-sample CI for .[Hint:
A state legislator wishes to survey residents of her district to see what proportion of the electorate is aware of her position on using state funds to pay for abortions.a. What sample size is necessary if the 95% CI for p is to have width of at most .10 irrespective of p?b. If the legislator has
A sample of 56 research cotton samples resulted in a sample average percentage elongation of 8.17 and a sample standard deviation of 1.42 (“An Apparent Relation Between the Spiral Angle , the Percent Elongation E1, and the Dimensions of the Cotton Fiber,” Textile Research J., 1978:407–410).
The article “An Evaluation of Football Helmets Under Impact Conditions” (Amer. J. Sports Medicine, 1984: 233–237) reports that when each football helmet in a random sample of 37 suspension-type helmets was subjected to a certain impact test, 24 showed damage. Let p denote the proportion of
A random sample of 487 nonsmoking women of normal weight (body mass index between 19.8 and 26.0) who had given birth at a large metropolitan medical center was selected (“The Effects of Cigarette Smoking and Gestational Weight Change on Birth Outcomes in Obese and NormalWeight Women,” Amer. J.
A random sample of 539 households from a certain midwestern city was selected, and it was determined that 133 of these households owned at least one firearm (“The Social Determinants of Gun Ownership: Self-Protection in an Urban Environment,” Criminology, 1997: 629–640). Using a 95%
The Associated Press (October 9, 2002) reported that in a survey of 4722 American youngsters aged 6 to 19, 15% were seriously overweight (a body mass index of at least 30; this index is a measure of weight relative to height). Calculate and interpret a confidence interval using a 99% confidence
The article “Ultimate Load Capacities of Expansion Anchor Bolts” (J. of Energy Engr., 1993: 139–158) gave the following summary data on shear strength (kip) for a sample of 3/8-in. anchor bolts: n 78, x 4.25, s 1.30. Calculate a lower confidence bound using a confidence level of 90% for
Exercise 1.13 gave a sample of ultimate tensile strength observations (ksi). Use the accompanying descriptive statistics output from MINITAB to calculate a 99% lower confidence bound for true average ultimate tensile strength, and interpret the result.N Mean Median TrMean StDev SE Mean 153 135.39
The charge-to-tap time (min) for a carbon steel in one type of open hearth furnace was determined for each heat in a sample of size 46, resulting in a sample mean time of 382.1 and a sample standard deviation of 31.5. Calculate a 95%upper confidence bound for true average charge-to-tap time.
A random sample of 110 lightning flashes in a certain region resulted in a sample average radar echo duration of .81 sec and a sample standard deviation of .34 sec(“Lightning Strikes to an Airplane in a Thunderstorm,” J. of Aircraft, 1984: 607–611). Calculate a 99% (two-sided) confidence
Consider the next 1000 95% CIs for that a statistical consultant will obtain for various clients. Suppose the data sets on which the intervals are based are selected independently of one another. How many of these 1000 intervals do you expect to capture the corresponding value of ? What is the
A random sample of n 15 heat pumps of a certain type yielded the following observations on lifetime (in years):
a. Under the same conditions as those leading to the interval(7.5), P[(X )/(/n) 1.645] .95. Use this to derive a one-sided interval for that has infinite width and provides a lower confidence bound on . What is this interval for the data in Exercise 5(a)?b. Generalize the result of part (a)
Let 1 0, 2 0, with 1 2 . Then Pz1 z2 1 a. Use this equation to derive a more general expression for a 100(1 )% CI for of which the interval (7.5) is a special case.b. Let .05 and 1 /4, 2 3/4. Does this result in a narrower or wider interval than the interval
On the basis of extensive tests, the yield point of a particular type of mild steel-reinforcing bar is known to be normally distributed with 100. The composition of the bar has been slightly modified, but the modification is not believed to have affected either the normality or the value of .a.
Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed with true standard deviation .75.a. Compute a 95% CI for the true average porosity of a certain seam if the average porosity for 20 specimens from the seam was 4.85.b. Compute a
A CI is desired for the true average stray-load loss (watts)for a certain type of induction motor when the line current is held at 10 amps for a speed of 1500 rpm. Assume that strayload loss is normally distributed with 3.0.a. Compute a 95% CI for when n 25 and x 58.3.b. Compute a 95% CI for
Suppose that a random sample of 50 bottles of a particular brand of cough syrup is selected and the alcohol content of each bottle is determined. Let denote the average alcohol content for the population of all bottles of the brand under study. Suppose that the resulting 95% confidence interval
Each of the following is a confidence interval for true average (i.e., population mean) resonance frequency (Hz) for all tennis rackets of a certain type:(114.4, 115.6) (114.1, 115.9)a. What is the value of the sample mean resonance frequency?b. Both intervals were calculated from the same sample
Consider a normal population distribution with the value of known.a. What is the confidence level for the interval x2.81/n?b. What is the confidence level for the interval x1.44/n?c. What value of z/2 in the CI formula (7.5) results in a confidence level of 99.7%?d. Answer the question posed in
6. Test the significance of r2 and R2 using ANOVA
5. Interpret SPSS multiple regression output
4. Calculate and interpret the coefficient of determination (r2) and Pearson’s correlation coefficient (r)
3. Calculate and interpret a and b
2. Construct and interpret straight-line graphs and best-fitting lines
1. Describe linear relationships and prediction rules for bivariate and multiple regression models
61. Analogous to a Latin square, a Greco–Latin square design can be used when it is suspected that three extraneous factors may affect the response variable and all four factors(the three extraneous ones and the one of interest) have the same number of levels. In a Latin square, each level of the
60. The article “Food Consumption and Energy Requirements of Captive Bald Eagles” (J. Wildlife Mgmt., 1982: 646–654)investigated mean gross daily energy intake (the response variable) for different diet types (factor A, with three levels)and temperature (factor B, with three levels). Summary
59. The bond strength when mounting an integrated circuit on a metalized glass substrate was studied as a function of factor A adhesive type, factor B cure time, and factor C conductor material (copper and nickel). The data follows, along with an ANOVA table from MINITAB. What conclusions can
57. One property of automobile air bags that contributes to their ability to absorb energy is the permeability (ft3/ft2/min) of the woven material used to construct the air bags. Understanding how permeability is influenced by various factors is important for increasing the effectiveness of air
56. Factorial designs have been used in forestry to assess the effects of various factors on the growth behavior of trees. In one such experiment, researchers thought that healthy spruce seedlings should bud sooner than diseased spruce seedlings(“Practical Analysis of Factorial Experiments in
55. Impurities in the form of iron oxides lower the economic value and usefulness of industrial minerals, such as kaolins, to ceramic and paper-processing industries. A 24 experiment was conducted to assess the effects of four factors on the percentage of iron removed from kaolin samples
54. Coal-fired power plants used in the electrical industry have gained increased public attention because of the environmental problems associated with solid wastes generated by large-scale combustion (“Fly Ash Binders in Stabilization of FGD Wastes,” J. of Environmental Engineering,
53. In an automated chemical coating process, the speed with which objects on a conveyor belt are passed through a chemical spray (belt speed), the amount of chemical sprayed (spray?
52. Four plots were available for an experiment to compare clover accumulation for four different sowing rates (“Performance of Overdrilled Red Clover with Different Sowing Rates and Initial Grazing Managements,” N. Zeal. J. Exp. Ag., 1984:71–81). Since the four plots had been grazed
50. The results of a study on the effectiveness of line drying on the smoothness of fabric were summarized in the article “LineDried vs. Machine-Dried Fabrics: Comparison of Appearance, Hand, and Consumer Acceptance” (Home Econ. Research J., 1984: 27–35). Smoothness scores were given for nine
49. A half-replicate of a 25 experiment to investigate the effects of heating time (A), quenching time (B), drawing time (C), position of heating coils (D), and measurement position (E)on hardness of steel castings resulted in the accompanying data. Construct the ANOVA table, and (assuming
48. Suppose that in the rocket thrust problem of Exercise 43, enough resources had been available for only a half-replicate of the 24 experiment.a. If the effect ABCD is chosen as the defining effect for the replicate and the group of eight treatments for which data is obtained includes treatment
47.a. In a seven-factor experiment (A, . . . , G), suppose a quarterreplicate is actually carried out. If the defining effects are ABCDE and CDEFG, what is the third nonestimable effect and what treatments are in the group containing(1)? What are the alias groups of the seven main effects?
45.a. An experiment was carried out to investigate the effects on audio sensitivity of varying resistance (A), two capacitances (B, C), and inductance of a coil (D) in part of a television circuit. If four blocks were used with four treatments per block, and the defining effects for confounding
44.a. In a 24 experiment, suppose two blocks are to be used, and it is decided to confound the ABCD interaction with the block effect. Which treatments should be carried out in the first block [the one containing the treatment (1)], and which treatments are allocated to the second block?b. In an
43. The article “Statistical Design and Analysis of Qualification Test Program for a Small Rocket Engine” (Industrial Quality Control, 1964: 14–18) presents data from an experiment to assess the effects of vibration (A), temperature cycling (B), altitude cycling (C), and temperature for
42. The following data on power consumption in electric furnace heats (kW consumed per ton of melted product) resulted from a 24 factorial experiment with three replicates(“Studies on a 10-cwt Arc Furnace,” J. Iron and Steel Institute, 1956: 22). The factors were nature of roof A (low, high),
41. In Exercise 39, suppose a low water temperature has been used to obtain the data. The entire experiment is then repeated with a higher water temperature to obtain the following
40. In a study of processes used to remove impurities from cellulose goods (“Optimization of Rope-Range Bleaching of Cellulosic Fabrics,” Textile Research J., 1976: 493–496), the following data resulted from a 24 experiment involving the desizing process. The four factors were enzyme
39. The accompanying data resulted from a 23 experiment with three replications per combination of treatments designed to study the effects of concentration of detergent (A), concentration of sodium carbonate (B), and concentration of sodium carboxymethyl cellulose (C) on cleaning ability of a
38. The accompanying data resulted from an experiment to study the nature of dependence of welding current on the three factors: welding voltage, wire feed speed, and tip-to-workpiece distance. There were two levels of each factor (a 23 experiment) with two replications per combination of levels
37. A four-factor ANOVA experiment was carried out to investigate the effects of fabric (A), type of exposure (B), level of exposure (C), and fabric direction (D) on extent of color change in exposed fabric as measured by a spectrocolorimeter. Two observations were made for each of the three
35. The article “Variation in Moisture and Ascorbic Acid Content from Leaf to Leaf and Plant to Plant in Turnip Greens”(Southern Cooperative Services Bull., 1951: 13–17) uses a Latin square design in which factor A is plant, factor B is leaf size (smallest to largest), factor C (in
34. The article “The Responsiveness of Food Sales to Shelf Space Requirements” (J. Marketing Research, 1964:63–67) reports the use of a Latin square design to investigate the effect of shelf space on food sales. The experiment was carried out over a 6-week period using six different stores,
32. When factors A and B are fixed but factor C is random and the restricted model is used (see the footnote on page 416; there is a technical complication with the unrestricted model here),
31. The article “Kolbe–Schmitt Carbonation of 2-Naphthol” (Industrial and Eng. Chemistry: Process and Design Development, 1969: 165–173) presented the accompanying data on percentage yield of BON acid as a function of reaction time (1, 2, and 3 hours), temperature (30, 70, and 100°C), and
29. The article “An Analysis of Variance Applied to Screw Machines” (Industrial Quality Control, 1956: 8–9) describes an experiment to investigate how the length of steel bars was affected by time of day (A), heat treatment applied (B), and screw machine used (C). The three times were 8:00
28. To see whether thrust force in drilling is affected by drilling speed (A), feed rate (B), or material used (C), an experiment using four speeds, three rates, and two materials was performed, with two samples (L 2) drilled at each combination of levels of the three factors. Sums of squares
25. Show how a 100(1 )% t CI for i i′ can be obtained.Then compute a 95% interval for 2 3 using the data from Exercise 19. [Hint: With 2 3, the result of Exercise 24a indicates how to obtain ˆ. Then compute V(ˆ) and ˆ, and obtain an estimate of ˆby using MSE to estimate
23. The accompanying data was obtained in an experiment to investigate whether compressive strength of concrete cylinders depends on the type of capping material used or variability in different batches (“The Effect of Type of Capping Material on the Compressive Strength of Concrete Cylinders,”
22. A study was carried out to compare the writing lifetimes of four premium brands of pens. It was thought that the writing surface might affect lifetime, so three different surfaces were randomly selected. A writing machine was used to ensure that conditions were otherwise homogeneous (e.g.,
20. The current (in A) necessary to produce a certain level of brightness of a television tube was measured for two different types of glass and three different types of phosphor, resulting in the accompanying data (“Fundamentals of Analysis of Variance,” Industrial Quality Control, 1956:
19. The accompanying data table gives observations on total acidity of coal samples of three different types, with determinations made using three different concentrations of ethanolic NaOH (“Chemistry of Brown Coals,” Australian J. Applied Science, 1958: 375–379).Type of Coal Morwell
18. The accompanying data resulted from an experiment to investigate whether yield from a certain chemical process depended either on the formulation of a particular input or on mixer speed.Speed 60 70 80 189.7 185.1 189.0 1 188.6 179.4 193.0 190.1 177.3 191.1 Formulation 165.1 161.7 163.3 2 165.9
17. The article “Towards Improving the Properties of Plaster Moulds and Castings” (J. Engr. Manuf., 1991: 265–269)describes several ANOVAs carried out to study how the amount of carbon fiber and sand additions affect various characteristics of the molding process. Here we give data on casting
15. The power curves of Figures 10.5 and 10.6 can be used to obtain P(type II error) for the F test in two-factor ANOVA. For fixed values of 1, 2, . . . , I, the quantity2 (J/I)2 i /2 is computed. Then the figure corresponding to 1 I 1 is entered on the horizontal axis at the value,
14. Use the fact that E(Xij) i j with i j 0 to show that E(Xi X) i, so that ˆi Xi X is an unbiased estimator for i.
11. The “residuals” from a two-factor ANOVA with Kij 1 are the quantities xij (ˆ ˆi ˆj). A normal probability plot of these residuals can be used as a plausibility check of the normality assumption. Construct such a plot for the data of Example 11.1 and comment.
10. The strength of concrete used in commercial construction tends to vary from one batch to another. Consequently, small test cylinders of concrete sampled from a batch are “cured”for periods up to about 28 days in temperature- and moisturecontrolled environments before strength measurements
9. The article “The Effects of a Pneumatic Stool and a OneLegged Stool on Lower Limb Joint Load and Muscular Activity During Sitting and Rising” (Ergonomics, 1993: 519–535)gives the accompanying data on the effort required of a subject to arise from four different types of stools (Borg
8. The paper “Exercise Thermoregulation and Hyperprolactinaemia” (Ergonomics, 2005: 1547–1557) discussed how various aspects of exercise capacity might depend on the temperature of the environment. The accompanying data on body mass loss (kg) after exercising on a semi-recumbent cycle
7. The article “Rate of Stuttering Adaptation Under Two ElectroShock Conditions” (Behavior Research Therapy, 1967: 49–54)gives adaptation scores for three different treatments: (1) no shock, (2) shock following each stuttered word, and (3) shock during each moment of stuttering. These
5. In an experiment to assess the effect of the angle of pull on the force required to cause separation in electrical connectors, four different angles (factor A) were used and each of a sample of five connectors (factor B) was pulled once at each angle (“A Mixed Model Factorial Experiment in
4. In an experiment to see whether the amount of coverage of light-blue interior latex paint depends either on the brand of paint or on the brand of roller used, 1 gallon of each of four brands of paint was applied using each of three brands of roller, resulting in the following data (number of
3. The article “Adiabatic Humidification of Air with Water in a Packed Tower” (Chem. Eng. Prog., 1952: 362–370) reports data on gas film heat transfer coefficient (Btu/hr ft2 on °F) as a function of gas rate (factor A) and liquid rate (factor B).B 1(190) 2(250) 3(300) 4(400)1(200) 200 226
2. Four different coatings are being considered for corrosion protection of metal pipe. The pipe will be buried in three different types of soil. To investigate whether the amount of corrosion depends either on the coating or on the type of soil, 12 pieces of pipe are selected. Each piece is coated
1. The number of miles of useful tread wear (in 1000s) was determined for tires of each of five different makes of subcompact car (factor A, with I 5) in combination with each of four different brands of radial tires (factor B, with J 4), resulting in IJ 20 observations. The values SSA

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