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introduction to probability statistics

Probability And Statistics For Engineering And The Sciences 8th Edition Jay L Devore, Roger Ellsbury - Solutions

Fitting the simple linear regression model to the observations on modulus of elasticity and flexural strength given in Exercise 15 of Section 12.2 resulted in when and for . x 5 60 yˆ 5 9.741, s 7.592, sYˆ 5 .179 x 5 40 Yˆ 5 .253 yˆ 5 x 5 y 5 n 5 27a. Explain why is larger when than when .b.
The probability of a type II error for the t test for can be computed in the same manner as it was computed for the t tests of Chapter 8.If the alternative value of b1 is denoted by , the value of is first calculated, then the appropriate set of curves in Appendix Table A.17 is entered on the
Verify that if each xi is multiplied by a positive constant c and each yi is multiplied by another positive constantd, the t statistic for testing versus is unchanged in value (the value of will change, which shows that the magnitude of is not by itself indicative of model utility).
a. Verify that by using the rules of expected value from Chapter 5.b. Use the rules of variance from Chapter 5 to verify the expression for given in this section.
Use the rules of expected value to show that is an unbiased estimator for b0 (assuming that is unbiased for b1).
Refer to the data on and emission rate given in Exercise 19.a. Does the simple linear regression model specify a useful relationship between the two rates? Use the appropriate test procedure to obtain information about the P-value, and then reach a conclusion at significance level .01.b. Compute a
Mist (airborne droplets or aerosols) is generated when metal-removing fluids are used in machining operations to cool and lubricate the tool and workpiece. Mist generation is a concern to OSHA, which has recently lowered substantially the workplace standard. The article “Variables Affecting Mist
How does lateral acceleration—side forces experienced in turns that are largely under driver control—affect nausea as perceived by bus passengers? The article “Motion Sickness in Public Road Transport: The Effect of Driver, Route, and Vehicle” (Ergonomics, 1999: 1646–1664) reported data
Refer to the Minitab output of Exercise 20, in which wet deposition and .a. Carry out the model utility test at level .01, using the rejection region approach.b. Repeat part (a) using the P-value approach.c. Suppose it had previously been believed that when wet deposition increases by .1 g N/m2,
Exercise 15 of Section 12.2 included Minitab output for a regression of flexural strength of concrete beams on modulus of elasticity.s 5 5.240 R-sq 5 97.5% R-sq(adj) 5 97.3%21.128 20.48 runoff 5 21.13 1 0.827 y 5 runoff volume (both in m3)x 5 rainfall volume sbˆ 1 gy2 i 5 66,224, gxi yi 5 703,192
Exercise 16 of Section 12.2 gave data on and . Use the accompanying Minitab output to decide whether there is a useful linear relationship between rainfall and runoff, and then calculate a confidence interval for the true average change in runoff volume associated with a 1 m3 increase in rainfall
During oil drilling operations, components of the drilling assembly may suffer from sulfide stress cracking. The article“Composition Optimization of High-Strength Steels for Sulfide Cracking Resistance Improvement” (Corrosion Science, 2009: 2878–2884) reported on a study in which the
Reconsider the situation described in Exercise 7, in which accelerated strength of concrete and cured strength.Suppose the simple linear regression model is valid for x between 1000 and 4000 and that and .Consider an experiment in which , and the x values at which observations are made are, and .a.
Consider the following three data sets, in which the variables of interest are and time. Based on a scatter plot and the values of s and r2, in which situation would simple linear regression be most(least) effective, and why
a. Consider the data in Exercise 20.Suppose that instead of the least squares line passing through the points, we wish the least squares line passing through . Construct a scatter plot of the points and then of the points. Use the plots to explain intuitively how the two least squares lines are
Suppose an investigator has data on the amount of shelf space x devoted to display of a particular product and sales revenue y for that product. The investigator may wish to fit a model for which the true regression line passes through (0, 0). The appropriate model is . Assume that are observed
Show that the “point of averages” lies on the estimated regression line.
The accompanying data was read from a graph that appeared in the article “Reactions on Painted Steel Under the Influence of Sodium Chloride, and Combinations Thereof”(Ind. Engr. Chem. Prod. Res. Dev., 1985: 375–378). The independent variable is SO2 deposition rate (mg/m2/d), and the dependent
a. Obtain SSE for the data in Exercise 19 from the defining formula , and compare to the value calculated from the computational formula.b. Calculate the value of total sum of squares. Does the simple linear regression model appear to do an effective job of explaining variation in emission rate?
Wrinkle recovery angle and tensile strength are the two most important characteristics for evaluating the performance of crosslinked cotton fabric. An increase in the degree of crosslinking, as determined by ester carboxyl band absorbence, improves the wrinkle resistance of the fabric (at the
A number of studies have shown lichens (certain plants composed of an alga and a fungus) to be excellent bioindicators of air pollution. The article “The Epiphytic Lichen Hypogymnia Physodes as a Biomonitor of Atmospheric Nitrogen and Sulphur Deposition in Norway” (Envir.Monitoring and
For the past decade, rubber powder has been used in asphalt cement to improve performance. The article “Experimental Study of Recycled Rubber-Filled High-Strength Concrete”(Magazine of Concrete Res., 2009: 549–556) includes a regression of on based on the following sample dataa. Obtain the
No-fines concrete, made from a uniformly graded coarse aggregate and a cement-water paste, is beneficial in areas prone to excessive rainfall because of its excellent drainage properties. The article “Pavement Thickness Design for NoFines Concrete Parking Lots,” J. of Trans. Engr.,
The article “Characterization of Highway Runoff in Austin, Texas, Area” (J. of Envir. Engr., 1998: 131–137) gave a scatter plot, along with the least squares line, of and for a particular location. The accompanying values were read from the plota. Does a scatter plot of the data support the
Values of modulus of elasticity (MOE, the ratio of stress, i.e., force per unit area, to strain, i.e., deformation per unit length, in GPa) and flexural strength (a measure of the ability to resist failure in bending, in MPa) were determined for a sample of concrete beams of a certain type,
The accompanying data on and appeared in the article“Plating of 60/40 Tin/Lead Solder for Head Termination Metallurgy” (Plating and Surface Finishing, Jan. 1997:y 5 rate of deposition(mm/min )x 5 current density (mA/cm2)gyi 2 5 17,454 gxi yi 5 25,825 gyi 5 346 gxi 2 5 39,095 n 5 14 gxi 5 517 y
Exercise 4 gave data on mass loading and mass removal. Values of relevant summary quantities area. Obtain the equation of the least squares line.b. Predict the value of BOD mass removal for a single observation made when BOD mass loading is 35, and calculate the value of the corresponding
Suppose that in a certain chemical process the reaction time y (hr) is related to the temperature (°F) in the chamber in which the reaction takes place according to the simple linear regression model with equation and.a. What is the expected change in reaction time for a 1°F increase in
Suppose the expected cost of a production run is related to the size of the run by the equation . Let Y denote an observation on the cost of a run. If the variables’size and cost are related according to the simple linear regression model, could it be the case that when and ? Explain.
Referring to Exercise 7, suppose that the standard deviation of the random deviation is 350 psi.a. What is the probability that the observed value of 28-day strength will exceed 5000 psi when the value of accelerated strength is 2000?b. Repeat part (a) with 2500 in place of 2000.c. Consider making
The article “Some Field Experience in the Use of an Accelerated Method in Estimating 28-Day Strength of Concrete” (J. of Amer. Concrete Institute, 1969: 895) consid ered regressing standard-cured strength (psi)against . Suppose the equation of the true regression line is .a. What is the
One factor in the development of tennis elbow, a malady that strikes fear in the hearts of all serious tennis players, is the impact-induced vibration of the racket-and-arm system at ball contact. It is well known that the likelihood of getting tennis elbow depends on various properties of the
The article “Objective Measurement of the Stretchability of Mozzarella Cheese” (J. of Texture Studies, 1992: 185–194)reported on an experiment to investigate how the behavior of mozzarella cheese varied with temperature. Consider the accompanying data on and(%) at failure of the cheese.
Bivariate data often arises from the use of two different techniques to measure the same quantity. As an example, the accompanying observations on hydrogen concentration(ppm) using a gas chromatography method and concentration using a new sensor method were read from a graph in the article “A New
The article “Exhaust Emissions from Four-Stroke Lawn Mower Engines” (J. of the Air and Water Mgmnt. Assoc., 1997: 945–952) reported data from a study in which both a baseline gasoline mixture and a reformulated gasoline were used. Consider the following observations on age (yr) and NOx
The efficiency ratio for a steel specimen immersed in a phosphating tank is the weight of the phosphate coating divided by the metal loss (both in mg/ft2). The article “Statistical Process Control of a Phosphate Coating Line” (Wire J. Intl., May 1997: 78–81) gave the accompanying data on tank
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 curve time, and factor C conductor material (copper and nickel). The data follows, along with an ANOVA table from Minitab. What conclusions can you
A chemical engineer has carried out an experiment to study the effects of the fixed factors of vat pressure (A), cooking time of pulp (B), and hardwood concentration (C) on the strength of paper. The experiment involved two pressures, four cooking times, three concentrations, and two observations
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 (“Factorial
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 volume), and the brand of chemical used(brand) are factors that may affect the uniformity of the coating applied. A
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. of Exp. Ag., 1984: 71–81). Since the four plots had been grazed
The water absorption of two types of mortar used to repair damaged cement was discussed in the article “Polymer Mortar Composite Matrices for Maintenance-Free, Highly Durable Ferrocement” (J. of Ferrocement, 1984: 337–345).Specimens of ordinary cement mortar (OCM) and polymer cement mortar
The results of a study on the effectiveness of line drying on the smoothness of fabric were summarized in the article“Line-Dried vs. Machine-Dried Fabrics: Comparison of Appearance, Hand, and Consumer Acceptance” (Home Econ. Research J., 1984: 27–35). Smoothness scores were given for nine
The article “Applying Design of Experiments to Improve a Laser Welding Process” (J. of Engr. Manufacture, 2008:1035–1042) included the results of a half replicate of a 24 experiment. The four factors were: A. Power (2900 W, 3300 W), B. Current (2400 mV, 3600 mV), C. Laterals cleaning(No,
a. In a seven-factor experiment , suppose a quarter-replicate 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?b. If the
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 were AB
The following data on power consumption in electricfurnace 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. of the Iron and Steel Institute, 1956: 22). The factors were nature of roof (A, low/high),
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 data. Use Yates’s algorithm on the entire set of 48 observations to obtain the sums of squares and ANOVA table, and then
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 the cleaning ability of a
The accompanying data resulted from an experiment to study the nature of dependence of welding current on three factors: welding voltage, wire feed speed, and tipto-workpiece distance. There were two levels of each factor (a 23 experiment) with two replications per combination of levels (the
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 fabrics,
The article “An Assessment of the Effects of Treatment, Time, and Heat on the Removal of Erasable Pen Marks from Cotton and Cotton/Polyester Blend Fabrics (J. of Testing and Eval., 1991: 394–397) reports the following sums of squares for the response variable degree of removal of marks:and .
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 parentheses)
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,
Because of potential variability in aging due to different castings and segments on the castings, a Latin square design with was used to investigate the effect of heat treatment on aging. With ,, summary statistics include and . Obtain the ANOVA table and test at level .05 the hypothesis that heat
When factors A and B are fixed but factor C is random and the restricted model is used (see the footnote on page 438;there is a technical complication with the unrestricted model here), anda. Based on these expected mean squares, what F ratios would you use to test for all i, j; and ?b. In an
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
The following summary quantities were computed from an experiment involving four levels of nitrogen (A), two times of planting (B), and two levels of potassium (C) (“Use and Misuse of Multiple Comparison Procedures,” Agronomy J., 1977: 205–208). Only one observation (N content, in percentage,
The article “An Analysis of Variance Applied to Screw Ma-chines” (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 A.M.,
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 drilled at each combination of levels of the three factors. Sums of squares were calculated
The output of a continuous extruding machine that coats steel pipe with plastic was studied as a function of the thermostat temperature profile (A, at three levels), the type of plastic (B, at three levels), and the speed of the rotating screw that forces the plastic through a tube-forming die (C,
When both factors are random in a two-way ANOVA experiment with K replications per combination of factor levels, the expected mean squares are, and.a. What F ratio is appropriate for testing versus ?b. Answer part (a) for testing versus and versus . HaB: sG H0B: sB 2 . 0 HaA: sA 2 5 0 2 . 0 H0A: sA
Show how a t CI for can be obtained. Then compute a 95% interval for using the data from Exercise 19.[Hint: With , the result of Exercise 24(a) indicates how to obtain . Then compute and and obtain an estimate of by using to estimate s (which identifies the appropriate number of df).]
a. Show that , so that is an unbiased estimator for ai (in the fixed effects model).b. With , show that is an unbiased estimator for (in the fixed effects model).
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,”
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., constant
In an experiment to investigate the effect of “cement factor”(number of sacks of cement per cubic yard) on flexural strength of the resulting concrete (“Studies of Flexural Strength of Concrete. Part 3: Effects of Variation in Testing Procedure,” Proceedings, ASTM, 1957: 1127–1139),
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)a. Assuming both effects to be
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.A statistical computer package gave , and .a. Does there appear to be interaction between the factors?b. Does
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
In an experiment to assess the effects of curing time (factor A)and type of mix (factor B) on the compressive strength of hardened cement cubes, three different curing times were used in combination with four different mixes, with three observations obtained for each of the 12 curing time–mix
The power curves of Figures 10.5 and 10.6 can be used to obtain (type II error) for the F test in two-factor ANOVA. For fixed values of , the quantity is computed. Then the figure corresponding to is entered on the horizontal axis at the value the power is read on the vertical axis from the curve
Use the fact that with to show that , so that is an unbiased estimator for
a. Show that a constant d can be added to (or subtracted from) each without affecting any of the ANOVA sums of squares.b. Suppose that each is multiplied by a nonzero constantc. How does this affect the ANOVA sums of squares?How does this affect the values of the F statistics FA and FB? What effect
Suppose that in the experiment described in Exercise 6 the five houses had actually been selected at random from among those of a certain age and size, so that factor B is random rather than fixed. Test versus using a level .01 test.
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 scale).
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 ergometer
The article “Rate of Stuttering Adaptation Under Two Electro-Shock 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
A particular county employs three assessors who are responsible for determining the value of residential property in the county. To see whether these assessors differ systematically in their assessments, 5 houses are selected, and each assessor is asked to determine the market value of each house.
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
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, one gallon of each of four brands of paint was applied using each of three brands of roller, resulting in the following data (number of
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).a. After constructing an ANOVA table, test at
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 ) in combination with each of four different brands of radial tires (factor B, with ), resulting in observations. The values, and were then computed. Assume
In Example 10.11, subtract from each observation in the ith sample to obtain a set of 18 residuals.Then construct a normal probability plot and comment on the plausibility of the normality assumption.
Suppose the ’s are “coded” by . How does the value of the F statistic computed from the ’s compare to the value computed from the ’s? Justify your assertion.
Four types of mortars—ordinary cement mortar (OCM), polymer impregnated mortar (PIM), resin mortar (RM), and polymer cement mortar (PCM)—were subjected to a compression test to measure strength (MPa). Three strength observations for each mortar type are given in the article“Polymer Mortar
Let be numbers satisfying . Then is called a contrast in the mi’s.Notice that with , which implies that every pairwise difference between mi’s is a contrast (so is, e.g., ).A method attributed to Scheffé gives simultaneous CIs with simultaneous confidence level for all possible contrasts (an
The critical flicker frequency (cff) is the highest frequency(in cycles/sec) at which a person can detect the flicker in a flickering light source. At frequencies above the cff, the light source appears to be continuous even though it is actually flickering. An investigation carried out to see
Four laboratories (1–4) are randomly selected from a large population, and each is asked to make three determinations of the percentage of methyl alcohol in specimens of a com- c1, c2,c, cI i 5 0 pound taken from a single batch. Based on the accompanying data, are differences among laboratories a
Refer to Exercise 38.What is b for the test when true average DNA content is identical for three of the diets and falls below this common value by 1 standard deviation (s) for the other two diets?
Referring to Exercise 38, construct a t CI for which measures the difference between the average DNA content for the starch diet and the combined average for the four other diets. Does the resulting interval include zero?
An article in the British scientific journal Nature (“Sucrose Induction of Hepatic Hyperplasia in the Rat,” August 25, 1972: 461) reports on an experiment in which each of five groups consisting of six rats was put on a diet with a different carbohydrate. At the conclusion of the experiment,
The article “Computer-Assisted Instruction Augmented with Planned Teacher/Student Contacts” (J. of Exp. Educ., Winter, 1980–1981: 120–126) compared five different methods for teaching descriptive statistics. The five methods were traditional lecture and discussion (L/D), programmed textbook
An experiment was carried out to compare flow rates for four different types of nozzle.a. Sample sizes were 5, 6, 7, and 6, respectively, and calculations gave . State and test the relevant hypotheses usingb. Analysis of the data using a statistical computer package yielded . At level .01, what
Simplify E(MSTr) for the random effects model when J . 1 5 J2 5 c5 JI 5 J g(mi) 5 mi(1 2 mi/n)Xij V(Xij) 5 si 2 5 npi(1 2 pi) 5 mi(1 2 mi m /n)
Suppose that is a binomial variable with parameters n and pi(so approximately normal when and ). Then nq npi $ 10 i $ 10 Xij since , .How should the ’s be transformed so as to stabilize the variance? [Hint: .]
Although tea is the world’s most widely consumed beverage after water, little is known about its nutritional value. Folacin is the only B vitamin present in any significant amount in tea, and recent advances in assay methods have made accurate determination of folacin content feasible. Consider
Samples of six different brands of diet/imitation margarine were analyzed to determine the level of physiologically active polyunsaturated fatty acids (PAPFUA, in percentages), resulting in the following data:Imperial 14.1 13.6 14.4 14.3 Parkay 12.8 12.5 13.4 13.0 12.3 Blue Bonnet 13.5 13.4 14.1
In an experiment to compare the quality of four different brands of magnetic recording tape, five 2400-ft reels of each brand (A–D) were selected and the number of flaws in each reel was determined.A: 10 5 12 14 8 B: 14 12 17 9 8 C: 13 18 10 15 18 D: 17 16 12 22 14 m2 5 m3, m1 5 m2 2 s m4 5 m2 1
When sample sizes are not equal, the noncentrality parameter is and . Referring to Exercise 22, what is the power of the test when, and ?

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