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inferential statistics
Probability And Statistics For Engineers And Scientists 9th Global Edition Ronald E. Walpole, Raymond Myers, Sharon L. Myers, Keying E. Ye - Solutions
13.9 Use Bartlett’s test at the 0.01 level of significance to test for homogeneity of variances in Exercise 13.5 on page 539.
13.8 Use Cochran’s test at the 0.01 level of significance to test for homogeneity of variances in Exercise 13.4 on page 539.
13.7 It has been shown that the fertilizer magnesium ammonium phosphate, MgNH4PO4 , is an effective supplier of the nutrients necessary for plant growth. The compounds supplied by this fertilizer are highly soluble in water, allowing the fertilizer to be applied directly on the soil surface or
13.6 The data in the following table represent the number of hours of relief provided by five different brands of headache tablets administered to 25 subjects experiencing fevers of 38◦C or more. Perform the analysis of variance and test the hypothesis at the 0.05 level of significance that the
13.5 The mitochondrial enzyme NADPH:NAD transhydrogenase of the common rat tapeworm (Hymenolepiasis diminuta) catalyzes hydrogen in the transfer from NADPH to NAD, producing NADH.This enzyme is known to serve a vital role in the tapeworm’s anaerobic metabolism, and it has recently been
13.4 Immobilization of free-ranging white-tailed deer by drugs allows researchers the opportunity to closely examine the deer and gather valuable physiological information.In the study Influence of Physical Restraint and Restraint Facilitating Drugs on Blood Measurements of White-Tailed Deer and
13.3 In an article “Shelf-Space Strategy in Retailing,”published in Proceedings: Southern Marketing Association, the effect of shelf height on the supermarket sales of canned dog food is investigated. An experiment was conducted at a small supermarket for a period of 8 days on the sales of a
13.2 A study measured the sorption (either absorption or adsorption) rates of three different types of organic chemical solvents. These solvents are used to clean industrial fabricated-metal parts and are potential hazardous waste. Independent samples from each type of solvent were tested, and
13.1 Six different machines are being considered for use in manufacturing rubber seals. The machines are being compared with respect to tensile strength of the product. A random sample of four seals from each machine is used to determine whether the mean tensile strength varies from machine to
12.72 Case Study: Consider the data set for Exercise 12.12, page 472 (hospital data), repeated here.Site x1 x2 x3 x4 x5 y 123456789 10 11 12 13 14 15 16 17 15.57 44.02 20.42 18.74 49.20 44.92 55.48 59.28 94.39 128.02 96.00 131.42 127.21 252.90 409.20 463.70 510.22 2463 2048 3940 6505 5723 11,520
12.71 Show that in choosing the so-called best subset model from a series of candidate models, choosing the model with the smallest s2 is equivalent to choosing the model with the smallest R2 adj .
12.70 A study was conducted to determine whether lifestyle changes could replace medication in reducing blood pressure among hypertensives. The factors considered were a healthy diet with an exercise program, the typical dosage of medication for hypertension, and no intervention. The pretreatment
12.69 An article in the Journal of Pharmaceutical Sciences (Vol. 80, 1991) presents data on the mole fraction solubility of a solute at a constant temperature.Also measured are the dispersion x1 and dipolar and hydrogen bonding solubility parameters x2 and x3 .A portion of the data is shown in the
12.68 A carbon dioxide (CO2 ) flooding technique is used to extract crude oil. The CO2 floods oil pockets and displaces the crude oil. In an experiment, flow tubes are dipped into sample oil pockets containing a known amount of oil. Using three different values of flow pressure and three different
12.67 Consider the data of Review Exercise 12.64.Suppose it is of interest to add some “interaction”terms. Namely, consider the model yi = β0 + β1x1i + β2x2i + β3x3i + β12x1ix2i+ β13x1ix3i + β23x2ix3i + β123x1ix2ix3i + i .(a) Do we still have orthogonality? Comment.(b) With the fitted
12.66 In a physiology exercise, an objective measure of aerobic fitness is the oxygen consumption in volume per unit body weight per unit time. Thirty-one individuals were used in an experiment to model oxygen consumption against their age in years (x1 ), weight in kilograms (x2 ), time to run
12.65 In a chemical engineering experiment dealing with heat transfer in a shallow fluidized bed, data are collected on the following four regressor variables: fluidizing gas flow rate, lb/hr (x1 ); supernatant gas flow rate, lb/hr (x2 ); supernatant gas inlet nozzle opening, millimeters (x3 ); and
12.64 A small experiment was conducted to fit a multiple regression equation relating the yield y to temperature x1 , reaction time x2 , and concentration of one of the reactants x3 . Two levels of each variable were chosen, and measurements corresponding to the coded independent variables were
12.63 Show that, in a multiple linear regression data set,n i=1 hii = p.
12.62 In the Department of Fisheries and Wildlife Conservation at Virginia Tech, an experiment was conducted to study the effect of stream characteristics on fish biomass. The regressor variables are as follows:average depth (of 50 cells), x1 ; area of in-stream cover(i.e., undercut banks, logs,
12.61 In an experiment to ascertain the effect of load, x, in lb/inches2 , on the probability of failure of specimens of a certain fabric type, an experiment was conducted in which numbers of specimens were exposed to loads ranging from 5 lb/in.2 to 90 lb/in.2. The numbers of “failures” were
12.60 From a set of streptonignic dose-response data, an experimenter desires to develop a relationship between the proportion of lymphoblasts sampled that contain aberrations and the dosage of streptonignic.Five dosage levels were applied to the rabbits used for the experiment. The data are as
12.59 In Exercise 12.28, page 482, we have the following data concerning wear of a bearing:y (wear) x1 (oil viscosity) x2 (load)193 230 172 91 113 125 1.6 15.5 22.0 43.0 33.0 40.0 851 816 1058 1201 1357 1115(a) The following model may be considered to describe the data:yi = β0 + β1x1i + β2x2i +
12.58 For Exercise 12.57, test H0: β1 = β6 = 0. Give P-values and comment.
12.57 The pull strength of a wire bond is an important characteristic. The following data give information on pull strength y, die height x1 , post height x2 , loop height x3, wire length x4 , bond width on the die x5 , and bond width on the post x6 . (From Myers, Montgomery, and Anderson-Cook,
12.56 In an effort to model executive compensation for the year 1979, 33 firms were selected, and data were gathered on compensation, sales, profits, and employment.Compensation, y Sales, x1 Profits, x2 Employ-Firm (thousands) (millions) (millions) ment, x3 123456789 450 387 368 277 676 454 507 496
12.55 Rayon whiteness is an important factor for scientists dealing in fabric quality. Whiteness is affected by pulp quality and other processing variables. Some of the variables include acid bath temperature, ◦C (x1 );cascade acid concentration, % (x2 ); water temperature,◦C (x3 ); sulfide
12.54 A client from the Department of Mechanical Engineering approached the Laboratory for Interdisciplinary Statistical Analysis at Virginia Tech for help in analyzing an experiment dealing with gas turbine engines. The voltage output of engines was measured at various combinations of blade speed
12.53 For the quadratic model of Exercise 12.51(b), give estimates of the variances and covariances of the estimates of β1 and β11 .
12.52 For the model of Exercise 12.50(a), test the hypothesis H0: β4 = 0, H1: β4 = 0.Use a P-value in your conclusion.
12.51 The following is a set of data for y, the amount of money (in thousands of dollars) contributed to the alumni association at Virginia Tech by the Class of 2005, and x, the number of years following graduation:y x y x 812.52 822.50 1211.50 1348.00 1301.00 2567.50 2526.50 123489 10 2755.00
12.50 For the punter data in Case Study 12.2, an additional response, “punting distance,” was also recorded. The average distance values for each of the 13 punters are given.(a) Using the distance data rather than the hang times, estimate a multiple linear regression model of the typeμY |x1
12.49 Use the techniques of backward elimination with α = 0.05 to choose a prediction equation for the data of Table 12.8.
12.48 For the data of Exercise 12.15 on page 472, use the techniques of(a) forward selection with a 0.05 level of significance to choose a linear regression model;(b) backward elimination with a 0.05 level of significance to choose a linear regression model;(c) stepwise regression with a 0.05 level
12.47 Consider the “hang time” punting data given in Case Study 12.2, using only the variables x2 and x3 .(a) Verify the regression equation shown on page 509.(b) Predict punter hang time for a punter with LLS =180 pounds and Power = 260 foot-pounds.(c) Construct a 95% confidence interval for
12.46 A study was done to determine whether the gender of the credit card holder was an important factor in generating profit for a certain credit card company. The variables considered were income, the number of family members, and the gender of the card holder. The data are as follows:Family
12.45 A study was done to assess the cost effectiveness of driving a four-door sedan instead of a van or an SUV (sports utility vehicle). The continuous variables are odometer reading and octane of the gasoline used.The response variable is miles per gallon. The data are presented here.MPG Car Type
12.44 For the data set given in Exericise 12.16 on page 473, can the response be explained adequately by any two regressor variables? Discuss.
12.43 Consider the data of Exercise 12.13 on page 472. Can the response, wear, be explained adequately by a single variable (either viscosity or load) in an SLR rather than with the full two-variable regression? Justify your answer thoroughly through tests of hypotheses as well as comparison of the
12.42 In Example 12.8, a case is made for eliminating x1 , powder temperature, from the model since the P-value based on the F-test is 0.2156 while P-values for x2 and x3 are near zero.(a) Reduce the model by eliminating x1 , thereby producing a full and a restricted (or reduced) model, and compare
12.41 Consider Example 12.3 on page 467. Compare the two competing models.First order: yi = β0 + β1x1i + β2x2i + i , Second order: yi = β0 + β1x1i + β2x2i+ β11x21 i + β22x22 i + β12x1ix2i + i .Use R2 adj in your comparison. Test H0 : β11 = β22 =β12 = 0. In addition, use the C.V.
12.40 Consider Example 12.4. Figure 12.1 on page 479 displays a SAS printout of an analysis of the model containing variables x1 , x2, and x3. Focus on the confidence interval of the mean response μY at the(x1, x2, x3 ) locations representing the 13 data points.Consider an item in the printout
12.39 Consider the data of Exercise 11.55 on page 457. Fit a regression model using weight and drive ratio as explanatory variables. Compare this model with the SLR (simple linear regression) model using weight alone. Use R2 , R2 adj, and any t-statistics (or F-statistics) you may need to compare
12.38 Consider the data for Exercise 12.36. Compute the following:R(β1 | β0 ), R(β1 | β0, β2, β3 ), R(β2 | β0, β1 ), R(β2 | β0, β1, β3 ), R(β3 | β0, β1, β2 ), R(β1, β2 | β3 ).Comment.
12.37 Consider the electric power data of Exercise 12.5 on page 470. Test H0 : β1 = β2 = 0, making use of R(β1, β2 | β3, β4 ). Give a P-value, and draw conclusions.
12.36 A small experiment was conducted to fit a multiple regression equation relating the yield y to temperature x1 , reaction time x2 , and concentration of one of the reactants x3 . Two levels of each variable were chosen, and measurements corresponding to the coded independent variables were
12.35 Repeat Exercise 12.17 on page 481 using an F-statistic.
12.34 For the model of Exercise 12.5 on page 470, test the hypothesis H0: β1 = β2 = 0, H1: β1 and β2 are not both zero.
12.33 Test whether the regression explained by the model in Exercise 12.5 on page 470 is significant at the 0.01 level of significance.
12.32 Test whether the regression explained by the model in Exercise 12.1 on page 470 is significant at the 0.01 level of significance.
12.31 Compute and interpret the coefficient of multiple determination for the variables of Exercise 12.1 on page 470.
12.30 Use the data from Exercise 12.16 on page 473.(a) Estimate σ2 using the multiple regression of y on x1 , x2, and x3 ,(b) Compute a 95% prediction interval for the observed gain with the three regressors at x1 = 15.0, x2 = 220.0, and x3 = 6.0.
12.29 Using the data from Exercise 12.28, test the following at the 0.05 level.(a) H0: β1 = 0 versus H1: β1 = 0;(b) H0: β2 = 0 versus H1: β2 = 0.(c) Do you have any reason to believe that the model in Exercise 12.28 should be changed? Why or why not?
12.28 Consider the following data from Exercise 12.13 on page 472.y (wear) x1 (oil viscosity) x2 (load)193 1.6 851 230 15.5 816 172 22.0 1058 91 43.0 1201 113 33.0 1357 125 40.0 1115 (a) Estimate σ2 using multiple regression of y on x1 and x2 .(b) Compute predicted values, a 95% confidence
12.27 Using the data of Exercise 12.5 on page 470 and the estimate of σ2 from Exercise 12.19, compute 95% confidence intervals for the predicted response and the mean response when x1 = 75, x2 = 24, x3 = 90, and x4 = 98.
12.26 For Exercise 12.8 on page 471, construct a 90%confidence interval for the mean compressive strength when the concentration is x = 19.5 and a quadratic model is used.
12.25 Using the data of Exercise 12.2 on page 470 and the estimate of σ2 from Exercise 12.17, compute 95% confidence intervals for the predicted response and the mean response when x1 = 900 and x2 = 1.00.
12.24 For the model of Exercise 12.1 on page 470, test the hypotheses that β1 = 2 against the alternative that β1 = 2. Use a P-value in your conclusion.
12.23 For the model of Exercise 12.2 on page 470, test the hypothesis that β1 = 0 at the 0.05 level of significance against the alternative that β1 = 0.
12.22 For the model of Exercise 12.7 on page 471, test the hypothesis that β2 = 0 at the 0.05 level of significance against the alternative that β2 = 0.
12.21 Referring to Exercise 12.5 on page 470, find the estimate of(a) σ2 b 2 ;(b) Cov(b1, b4 ).
12.20 Obtain estimates of the variances and the covariance of the estimators b1 and b2 of Exercise 12.2 on page 470.
12.19 For the data of Exercise 12.5 on page 470, estimateσ2 .
12.18 For the data of Exercise 12.1 on page 470, estimateσ2 .
12.17 For the data of Exercise 12.2 on page 470, estimateσ2 .
12.16 An engineer at a semiconductor company wants to model the relationship between the gain or hFE of a device (y) and three parameters: emitter-RS(x1 ), base-RS (x2 ), and emitter-to-base-RS (x3 ). The data are shown below:x1 , x2 , x3 , y, Emitter-RS Base-RS E-B-RS hFE 14.62 15.63 14.62 15.00
12.15 The personnel department of a certain industrial firm used 12 subjects in a study to determine the relationship between job performance rating (y) and scores on four tests. The data are as follows:y x1 x2 x3 x4 11.2 56.5 71.0 38.5 43.0 14.5 59.5 72.5 38.2 44.8 17.2 69.2 76.0 42.5 49.0 17.8
12.14 Eleven student teachers took part in an evaluation program designed to measure teacher effectiveness and determine what factors are important. The response measure was a quantitative evaluation of the teacher. The regressor variables were scores on four standardized tests given to each
12.13 A study was performed on a type of bearing to find the relationship of amount of wear y to x1 = oil viscosity and x2 = load. The following data were obtained. (From Response Surface Methodology, Myers, Montgomery, and Anderson-Cook, 2009.)y x1 x2 y x1 x2 193 172 113 1.6 22.0 33.0 851 1058
12.11 An experiment was conducted to study the size of squid eaten by sharks and tuna. The regressor variables are characteristics of the beaks of the squid. The data are given as follows:x1 x2 x3 x4 x5 y 1.31 1.55 0.99 0.99 1.01 1.09 1.08 1.27 0.99 1.34 1.30 1.33 1.86 1.58 1.97 1.80 1.75 1.72 1.68
12.10 The following data are given:x 0 1 2 3 4 5 6 y 1 4 5 3 2 3 6(a) Fit the cubic model μY |x = β0 +β1x+β2x2 +β3x3 .(b) Predict Y when x = 2.
12.9 (a) Fit a multiple regression equation of the form μY |x = β0 + β1x1 + β2x2 to the data of Example 11.8 on page 440.(b) Estimate the yield of the chemical reaction for a temperature of 225◦C.
12.8 The following is a set of coded experimental data on the compressive strength of a particular alloy at various values of the concentration of some additive:Concentration, Compressive x Strength, y 10.0 15.0 20.0 25.0 30.0 25.2 29.8 31.2 31.7 29.4 27.3 31.1 32.6 30.1 30.8 28.7 27.8 29.7 32.3
12.7 An experiment was conducted in order to determine if cerebral blood flow in human beings can be predicted from arterial oxygen tension (millimeters of mercury). Fifteen patients participated in the study, and the following data were collected:Blood Flow, Arterial Oxygen y Tension, x 84.33
12.6 An experiment was conducted on a new model of a particular make of automobile to determine the stopping distance at various speeds. The following data were recorded.Speed, v (km/hr) 35 50 65 80 95 110 Stopping Distance, d (m) 16 26 41 62 88 119(a) Fit a multiple regression curve of the form
12.5 The electric power consumed each month by a chemical plant is thought to be related to the average ambient temperature x1 , the number of days in the month x2 , the average product purity x3, and the tons of product produced x4 . The past year’s historical data are available and are
12.4 An experiment was conducted to determine if the weight of an animal can be predicted after a given period of time on the basis of the initial weight of the animal and the amount of feed that was eaten. The following data, measured in kilograms, were recorded:Final Initial Feed Weight, y
12.3 Suppose in Review Exercise 11.53 on page 457 that we were also given the number of class periods missed by the 12 students taking the chemistry course.The complete data are shown.Chemistry Test Classes Student Grade, y Score, x1 Missed, x2 123456789 10 11 12 85 74 76 90 85 87 94 98 81 91 76 74
12.2 In Applied Spectroscopy, the infrared reflectance spectra properties of a viscous liquid used in the electronics industry as a lubricant were studied. The designed experiment consisted of the effect of band frequency x1 and film thickness x2 on optical density y using a Perkin-Elmer Model 621
12.1 A set of experimental runs was made to determine a way of predicting cooking time y at various values of oven width x1 and flue temperature x2. The coded data were recorded as follows:y x1 x2 6.40 15.05 18.75 30.25 44.85 48.94 51.55 61.50 100.44 111.42 1.32 2.69 3.56 4.41 5.35 6.20 7.12 8.87
11.68 Project: Project: This project can be done in groups or as individuals. Each group or person must find a set of data, preferably but not restricted to their field of study. The data needs to fit the regression framework with a regression variable, x, and a response variable, y. Carefully make
11.67 Consider the fictitious set of data shown below, where the line through the data is the fitted simple linear regression line. Sketch a residual plot. x
11.66 Show the necessary steps in converting the equation r = b 1 s/√Sx x to the equivalent form t = r√√ n−2 1−r 2 .
11.65 Suppose that an experimenter postulates a model of the type, Yi = β0 +β1x1i+i, i= 1, 2, . . . , n, when, in fact, an additional variable (say, x2 independent of x1 ) also contributes linearly to the response.The true model is then given by: Yi = β0 + β1x1i +β2x2i + i, i= 1, 2, . . . ,
11.64 In Review Exercise 11.62, the student was required to show thatn i=1(yi − ˆyi) = 0 for a standard simple linear regression model. Does the same hold for a model with zero intercept? Show why or why not.
11.63 Consider the situation presented in Exercise 11.62 but suppose that n = 2 (that is, only two data points are available). Prove that the coefficient of correlation is +1 or −1 as the regression line increases or decreases.
11.62 Show, in the case of a least squares fit to the simple linear regression model Yi = β0 + β1xi + i, i= 1, 2, . . . , n, thatn i=1(yi − ˆyi) =n i=1 ei = 0.
11.61 For a simple linear regression model Yi = β0 + β1xi + i, i= 1, 2, . . . , n, where the i are independent and normally distributed with zero means and equal variances σ2 , show that ¯ Y and B1 =n i=1(xi − ¯x)Yin i=1(xi − ¯x)2 have zero covariance.
11.60 Assuming that the i are independent and normally distributed with zero means and common varianceσ2 , show that B0 , the least squares estimator ofβ0 in μY |x = β0 + β1x, is normally distributed with mean β0 and variance σ2B 0 = n i=1 x2i n n i=1 (xi − ¯x)2 σ2 .
11.59 For the simple linear regression model, prove that E(s2) = σ2 .
11.58 Suppose a scientist postulates a model:Yi = 5+β1xi + i, i = 1, 2, . . . , n.(a) What is the appropriate least squares estimator ofβ1 ? Justify your answer.(b) What is the variance of the slope estimator?
11.57 Physical fitness testing is an important aspect of athletic training. A common measure of the magnitude of cardiovascular fitness is the maximum volume of oxygen uptake during strenuous exercise. A study was conducted on 24 middle-aged men to determine the influence on oxygen uptake of the
11.56 Observations on the yield of a chemical reaction taken at various temperatures were recorded as follows:x (◦C) y (%) x (◦C) y (%)150 75.4 150 77.7 150 81.2 200 84.4 200 85.5 200 85.7 250 89.0 250 89.4 250 90.5 300 94.8 300 96.7 300 95.3(a) Plot the data.(b) Does it appear from the plot as
11.55 Consider the vehicle data from Consumer Reports in Figure 11.30 on page 460. Weight is in tons, mileage in miles per gallon, and drive ratio is also indicated.A regression model was fitted relating weight x to mileage y. A partial SAS printout in Figure 11.30 on page 460 shows some of the
11.54 The business section of the Washington Times in March of 1997 listed 21 different used computers and printers and their sale prices. Also listed was the average hover bid. Partial results from regression analysis using SAS software are shown in Figure 11.29 on page 459.(a) Explain the
11.53 The following data represent the chemistry grades for a random sample of 12 freshmen at a certain college along with their scores on an intelligence test administered while they were still seniors in high school.Test Chemistry Student Score, x Grade, y 123456789 10 11 12 65 50 55 65 55 70 65
11.52 An experiment was designed for the Department of Materials Science and Engineering at Virginia Tech to study hydrogen embrittlement properties based on electrolytic hydrogen pressure measurements.The solution used was 0.1 N NaOH, and the material was a certain type of stainless steel. The
11.51 With reference to Exercise 11.9 on page 419, construct(a) a 95% confidence interval for the average weekly sales when $45 is spent on advertising;(b) a 95% prediction interval for the weekly sales when$45 is spent on advertising.
11.50 The number of cigarettes consumed per day and the rate of coronary heart disease per 10000 people of the population is recorded for two states.Cases of coronary heart Cigarettes disease (per 10000)per day State I State II 9 21 24 8 19 13 7 16 14 6 11 20 5 15 13 4 11 6 3 4 11(a) Estimate the
11.49 The Laboratory for Interdisciplinary Statistical Analysis at Virginia Tech analyzed data on normal woodchucks for the Department of Veterinary Medicine. The variables of interest were body weight in grams and heart weight in grams. It was desired to develop a linear regression equation in
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