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Statistics And Probability With Applications For Engineers And Scientists Using MINITAB R And JMP 2nd Edition Irwin Guttman, Kalanka P. Jayalath, Bhisham C. Gupta - Solutions
11. Refer to Problem 7 of Section 16.3. Assuming normality of the Y’s,(a) Use the relevant t-statistic to test each of the hypotheses H0 : βi = 0 versus H1 :βi= 0, i = 2, 3, 4, 5. Use α = 0.05.(b) Find a 95% confidence interval for each of the βi, i = 2, 3, 4, 5.
10. Repeat Problem 6 above for the model in Problem 7(a) of Section 16.3.
9. For the model in Problem 5, construct the ANOVA table and then use this ANOVA table to determine the value of R2 and R2 adj .Review Practice Problems 749
8. Refer to Problem 5 above. Assuming normality of the Y’s,(a) Use a t-statistic to test each of the hypotheses at the 5% level of significance:H0 : βi = 0 versus H1 : βi= 0, i= 1, 3, 4, 6(b) Find a 95% confidence interval for each of the βi, i = 1, 3, 4, 6.
7. Use the results of Problems 5(b) and 6, to find the variance and covariance matrix of the estimators ˆ β3 and ˆ β4 in the model E(Y ) = β0 + β1X1 + β3X3 + β4X4 + β6X6
6. Find the inverse matrix (XX)−1 for the model in Problem 5.
5. Use the data in Problem 7 of Section 16.3 to(a) Fit a multiple linear regression model using X1, X3, X4, and X6 as the predictor variables;(b) Estimate the variance σ2.
4. In Problem 4 of Section 16.3, find a 95% prediction interval for Y when X1 = 75, X2 = 70, assuming that εi’s are normally distributed with mean zero and variance σ2.
3. In Problem 2, let H = X(XX)−1X be the HAT matrix for the general regression model Yi = β0 + β1Xi1 + β2Xi2 + · · · + βkXik + εi, i= 1, 2, . . . , n Show that the HAT matrix H is idempotent. If I is an identity matrix, then show that (I − H) is also an idempotent matrix.
2. In Problem 1, what are the dimensions of the HAT matrix H = X(XX)−1X?
1. Consider the multiple linear regression model in four predictor variables, that is, Yi = β0 + β1Xi1 + β2Xi2 + β3Xi3 + β4Xi4 + εi, i= 1, 2, . . . , 10 Use matrix notation to describe this model. Define the least-squares normal equations for this model. Assuming that X is a full-rank matrix,
An engineer at a semiconductor company wantsto model the relationship between the device gain or hFE (Y) and the three parameters(independent variables): emitter-RS (X1), base-RS (X2), and emitter-to-base-RS (X3). Thedata are shown below:X1 emitter-RS X2 base-RS X3 B-E-RS Y hFE-1M-5V14.620 226.00
3. A manager of a manufacturing company conducted a study to investigate the relationship between the years X of service of an engineer and his/her performance(Y = 1 indicating satisfactory and Y = 0 not satisfactory). The following data give the results for 15 randomly selected engineers who work
2. A travel agency conducted a study to investigate the relationship between middle-class families’ household income in thousands of dollars X and traveling at least 500 miles for vacation. Ten middle-class families were randomly selected and their family income and status (Y = 1 implies traveled
1. The following data give MCAT scores X of 10 applicants who apply for admission to medical school. The result of each applicant is either accepted (Y = 1) or not accepted (Y = 0). Fit a logistic regression model and interpret your results.Applicant 1 2 3 4 5 6 7 8 9 10 X 28 32 29 33 27 31 30 34
8. Refer to Problem 3 above. Determine a 95% confidence interval for E(Y |X0) and a prediction interval for Y when X0 = (50, 1) and X0 = (50, 0).
7. Refer to Problem 5 above. Develop the ANOVA table for the model considered in Problem 5, and use an appropriate F-test to evaluate whether or not the model fitted is appropriate. Use α = 0.01.
6. Refer to Problem 5 above. Determine a 95% confidence interval for E(Y |X0) and a prediction interval for Y when X0 = (25, 4, 3, 2,Brick).
5. Refer to Table 16.4.3 in Example 16.4.2. As an addendum to the data of Table 16.4.3, incorporate an additional qualitative predictor variable to be called“house siding” having four categories, namely Wood, Vinyl, Stucco, and Brick.Specifically, the first seven homes used wood siding, next
4. Refer to Problem 3 above. Develop the ANOVA table for the model you considered in Problem 3, and use an appropriate F-test to evaluate whether or not the model you fitted is appropriate. Use α = 0.05.726 16 Multiple Linear Regression Analysis
3. An educator is interested in studying the difference between public and private four-year institutions that award a chemical engineering degree. The educator selected eight public institutions and eight private institutions. The following data give the number of graduates (Y) hired during campus
2. Refer to Problem 1 above. Suggest to the CEO how she can use this model to achieve the company’s goal.
1. A company has two water testing labs, Lab A and Lab B. Lab A is fully equipped with modern facilities, whereas Lab B does not have all those facilities. The following data give the number of water samples X1 arriving per day in each of the two labs and the total of technician hours Y taken by
11. Runs were made at various conditions of saturation X1 and transisomers X2. The response SCI, denoted by Y, is given below for the corresponding conditions of X1 and X2.(a) Fit the regression model E(Y ) = β0 + β1X1 + β2X2 to the data below.(b) Test the hypothesis H0 : β1 = β2 = 0 versus H1
10. Refer to Problem 7 part (c). Find a 95% confidence interval for E(Y ) and a 95%prediction interval for Y at X2 = 20, X3 = 30, X4 = 90, and X5 = 2.0, assuming normality of the Y’s.
9. Refer to Problem 8.(a) For the model in Problem 8, and assuming normality of the Y’s, test the hypothesis H0 : β4 = β5 = β6 = 0 versus H1 : Not all parameters β4, β5, β6 are zero. Use α = 0.05.(b) Find the p-value for the test in part (a).
8. Fit a multiple linear regression model Y = β0 + β1X1 + β2X2 + · · · + β6X6 + ε, to the data in Problem 7.
7. The pull strength of a wire bond is an important characteristic. The table below gives 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 and Montgomery, 1995, used with permission).(a)
6. In Problem 4 find:(a) An estimate of the variance σ2. (Hint: ˆσ2 = MSE = SSE/(n − 3).)(b) A 95% confidence interval for E(Y |X1 = 75,X2 =70), assuming that εis are normally distributed with mean zero and variance σ2.
5. (a) Use the least square normal equations in Problem 4 to estimate the regression coefficients β0, β1, β2 for the regression plane E(Y |X1,X2) = β0 + β1X1 + β2X2.(b) Use the result obtained in (a) to estimate the value of Y if a piece of yarn is such that X1 = 75 and X2 = 70.
4. A study was made of the relationship among skein strength (Y, in lb) of #225 cotton yarn, mean fiber length (X1, in 0.01 in.) and fiber tensile strength (X2, in 1000 psi). Twenty combinations of X1 and X2 values were used and Y observed at each of these combinations. The observations yield the
3. In Problem 1, determine the normal equations and find the least square estimators for the regression coefficients β0, β1, β2, and β3.
2. (a) Set up a second-order interaction multiple linear regression model in three predictor variables.(b) Set up a second-order complete multiple linear regression model in three predictor variables.(c) Set up the models in (a) and (b) using matrix notation.
1. Set up a first-order multiple linear regression model in three predictor variables.
38. Refer to Problem 35.(a) Determine all the residuals in Problem 35 and test if the normality assumption is satisfied.(b) Plot the residuals versus values of ˆ Y . Comment on the assumption of constant variance of the Yi.(c) Construct an appropriate graph to check the assumption of independence
37. Refer to Problem 35.(a) Fit a simple regression model using log Y as the dependent variable.(b) Estimate the value of Y at X = 1.50, using the model fitted in (a) and the model fitted in Problem 35. Compare the two results and comment on them.
36. Refer to Problem 35.(a) Estimate σ2 and the standard error of ˆ β0 and ˆ β1.(b) Find 95% confidence intervals for β0 and β1.692 15 Simple Linear Regression Analysis(c) Determine the sample correlation coefficient between X and Y.(d) Test the hypothesis H0 : ρ = 0 versus H1 : ρ = 0.
35. Thirteen specimens of 90/10 Cu–Ni alloys, each with a specific iron content, were tested in a corrosion-wheel setup. The wheel was rotated in seawater at 30 ft/s for 60 days. The corrosion was measured in weight loss in milligrams/square decimeter/day, MDD. The data collected are given below
34. Refer to Problem 3, Section 15.2. Test the hypothesis H0 : ρ = 0 versus H1 : ρ < 0.Use α = 0.01. Find the observed level of significance (p-value). Give the practical interpretation of the p-value.
33. Refer to Problem 3, Section 15.2.(a) Determine the correlation coefficient between X and Y for the data in Problem 3.(b) Test the hypothesis H0 : ρ = 0 versus H1 : ρ > 0. Use α = 0.05 Find the observed level of significance (p-value). Give the practical interpretation of the p-value.
32. The following data give 12 results of measuring the thickness Y of the silver film deposited when an amount X of a certain acid mix is used in a process. The values of X were preselected, and the corresponding values of Y are listed below in the order obtained:X 4 6 2 5 7 6 3 8 5 3 1 5 Y 197
31. The following data are 28 observations Y on the yield of a certain by-product when certain temperatures X ◦C are used in a chemical process:X 22 30 28 30 48 29 50 39 47 39 30 15 42 31 Y 72 91 69 81 71 85 79 77 81 75 73 62 70 65 X 28 10 3 12 19 33 27 4 27 36 46 12 17 8 Y 62 57 58 62 68 71 60
30. In Problem 29, after concluding which model is the “better fit,” find a 99% prediction interval for Y and a 99% confidence interval for E(Y |X) at X = 22 and 35, using the model with the better fit.
29. Consider the following data set.X 6 8 11 15 17 20 25 30 33 40 Y 20 23 32 38 40 44 39 32 26 24(a) Fit these data to the following models:Yi = β0 + β1Xi + εi, i= 1, . . . , 10 Yi = τ0 + τ1Xi + τ2X2 i + εi, i= 1, . . . , 10(b) Perform residual analysis for both fits and conclude which is a
28. Refer to Problem 27. Assuming that a linear regression model is appropriate.(a) Find a 95% prediction interval for Y when X = 150 mg.(b) Find a 95% confidence interval for E(Y |X) when X = 150 mg.(c) Compare the confidence interval in (b) with the prediction interval in (a) and indicate which
27. A new cholesterol-lowering drug is being tested on eight randomly chosen patients.Since the appropriate dose of the drug is yet unknown, the chosen doses are varied and the resulting level of cholesterol changes in each patient are as given below:Dose (mg), X 190 175 100 125 140 190 200 175
26. In Problem 24, use the log transformation on both the X and Y variables and fit the least-square line to the transformed data. Perform the residual analysis in this problem and in Problem 24 and compare the two fits. Give your conclusion about which fit is the better one. State the model that
25. Refer to Problem 24, find the correlation coefficient between the X and Y variables.Perform a test, using the 5% level of significance, of the hypothesis H0 : ρ = 0 versus H1 : ρ > 0.690 15 Simple Linear Regression Analysis
24. The manager of a manufacturing company believes that experience is the most valuable variable in determining a worker’s productivity. She collects data on productivity of 10 workers with known number of years of experience. The data collected are given below:Experience, X 15 15 13 20 13 20 11
23. In Problem 16, perform the residual analysis. Do you find any abnormalities? If you do find any abnormalities, then suggest some remedies so that a suitable model can be fitted.
22. In Problem 17, construct the ANOVA table and perform the F-test to learn whether the assumption that a linear regression model is the true model; do this at the 5%level of significance. State your conclusion.
21. In Problem 20, discuss whether it is reasonable to find a confidence interval for the observation Y or its expected value E(Y |X) when X = 11.
20. In Problem 13, find a 99% confidence interval for the observation Y and its expected value E(Y |X = x), at x = 5.50 and x = 8.20.
19. In Problem 18, find a 95% confidence interval for the slope parameter τ .
18. Estimate the acceleration, say τ , of a body disturbed from rest by a constant applied force. The approximate model is V = τt, where V is the velocity at time t.t (s) 1 2 3 4 5 6 7 8 Velocity V (ft/s) 34.2 57.6 94.3 121.0 146.4 175.2 212.8 247
17. Sulfur dioxide can be removed from flue gases at low temperatures (approximately 600◦F) through the use of a dry absorbent, alkalized alumina. The absorbent, when spent, is later regenerated in a separate process with elemental sulfur produced as a by-product. One series of experiments
16. In a study of internal combustion engines, the data given were observed with Y (in units of BTU/lb), the net work provided by a cylinder, as a function of the fuel fraction X, where 0 < X ≤ 1.Fuel fraction 0.15 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.0 Net work 120 165 204 238 296 373
15. In an investigation of pure copper bars of a small diameter, the following shear stress and shear strain data were collected.Shear strain (%) 8.8 9.3 10.4 11.2 12.3 13.0 13.8 14.4 15.8 16.7 17.3 Shear stress (1000 psi) 11.8 11.9 11.4 11.6 12.3 12.7 13.3 13.7 13.8 14.4 14.5(a) Fit a
14. Experiments have shown that when a clean tungsten surface is heated by laser irradiation using a focused ruby laser, the rate of evaporation of tungsten from the surface is similar to that obtained by more conventional surface-heating methods. The following experimental data relates the
13. Test pieces of boiler plate undergo tests at various times during a production process.The measurements made are X, the force applied in tons per square inch at the time of removal from the process, and the resulting elongation Y of the test piece. For the results given below on 10 test pieces,
12. Repeat the instructions of Problem 10 for the data given below. These data (Bullis and Alderton) were obtained by examining the alpha resin content of six different specimens of hops by taking colorimeter readings X and by direct determination of the concentration (Y, in mg per 100 ml).Specimen
11. Repeat the instructions of Problem 10 for the data given below. The data set was obtained by measuring the tensile strength (Y) in 1000 psi and the Brinell Review Practice Problems 687 hardness (X) of each of 15 specimens of cold-drawn copper (data from Bowker and Lieberman, 1959).Specimen # X
10. Specimens of blood from 10 different animals were analyzed for blood count, say, Y (in units of 100) and packed cell volume count X (in mm3) with results as given below. Assuming normality, test the hypothesis that the true correlation coefficient ρbetween blood count and volume count is zero.
9. The effect of temperature (X in Kelvin) on the color (Y, coded units) of a product was investigated and the results obtained are given below. Construct an analysis of variance table for these data and then use it to test the hypothesis H0 : β1 = 0. If the test rejects H0, then fit these data to
8. The moisture X of the wet mix of a product is considered to have an effect on the density Y of the finished product. The moisture of the mix was controlled and the finished product densities were as shown below. Repeat the instructions of Problem 6 for this set of data, choosing convenient
7. The relationship (assumed linear) between the yield of bourbon Y and aging time X was studied by observing yields Yi from batches that have been allowed to age Xi = 2i years, i = 1, 2, . . . , 6. The results are given below. Repeat the instructions of Problem 6 for this set of data, but use X0 =
6. A study was made on the effect of pressure X on the yield Y of paint made by a certain chemical process. The results (in coded units) are given below. Repeat the instructions of Problem 3. In the notation of Problem 3 above, let X0 take the values−5, −3, −1, 1, 3, and 5 and draw the
5. An investigation of the (assumed) linear relationship between the load X on a spring and the subsequent length of the spring Y has been carried out with the results given below. Repeat the instructions of Problem 3 for this set of data.X 5 10 15 20 25 30 Y 7.25 8.12 8.95 9.90 10.9 11.8
4. A chemical engineer wants to fit a straight-line to the data found observing the tensile strength, Y, of 10 test pieces of plastic that have undergone baking (at a uniform temperature) for X minutes, where 10 values of X were preselected. The data(in coded units) is given below. Repeat the
3. An experiment is planned in which three observations will be taken at each of four temperatures, 30◦, 50◦, 70◦, and 90◦. When the experiment is actually done, the results obtained are those given below. Find, assuming η = E(Y |X) = β0 + β1X, the least-squares estimates of β0 and β1,
2. It is decided to measure the resistance of sheets of a certain metal at temperatures X of 100, 200, 300, 400, and 500 K. The resistances Y are found to be 4.7, 7.4, 12.4, 16.5, and 19.8, respectively. If the regression of Y on X is assumed to be linear, state the normal equations for the
1. Suppose that it has been assumed on prior theoretical grounds that E(Y |X) = η =β1X. To gain knowledge of β1, it is decided to experiment by setting the independent Review Practice Problems 685 variable X to each of the values x1, . . . , xn and to observe the resulting Y’s, say Y1, . . . ,
10. Refer to Problem 9.(a) Construct the ANOVA table for the data in Problem 9.(b) Use the ANOVA table in (a) to evaluate the fitted model in Problem 9. Useα = 0.01.(c) Determine the observed level of significance (p-value) in (b).(d) Calculate the coefficient of determination R2. Give the
9. Refer to Problem 8.(a) Construct the ANOVA table for the data in Problem 8.(b) Use the ANOVA table in part (a) to evaluate the fitted model in Problem 8.Use α = 0.01.(c) Determine the observed level of significance (p-value) in (b).(d) Calculate the coefficient of determination R2. Give the
8. Refer to Problem 7.(a) Construct the ANOVA table for the data in Problem 7.(b) Use the ANOVA table in (a) to evaluate the fitted model in Problem 7. Useα = 0.01.(c) Determine the observed level of significance (p-value) in (b).(d) Calculate the coefficient of determination R2. Give the
7. Refer to Problem 3.(a) Construct the ANOVA table for the data in Problem 3.(b) Use the ANOVA table in part (a) to evaluate the fitted model in Problem 3.Use α = 0.05.(c) Determine the observed level of significance (p-value) in (b).(d) Calculate the coefficient of determination R2. Give the
6. Refer to Problem 2.(a) Construct the ANOVA table for the data in Problem 2.(b) Use the ANOVA table in (a) to evaluate the fitted model in Problem 2. Useα = 0.05.(c) Determine the observed level of significance (p-value) in (b).(d) Calculate the coefficient of determination R2. Give the
5. Refer to Problem 1.(a) Construct the ANOVA table for the data in Problem 1.(b) Use the ANOVA table in part (a) to evaluate the fitted model in Problem 1.Use α = 0.05.(c) Determine the observed level of significance (p-value) in (b).(d) Calculate the coefficient of determination R2. Give the
4. The following data give the methyl mercury intake X and whole blood mercury Y in 12 subjects exposed to methyl mercury by eating contaminated fish. Assume 664 15 Simple Linear Regression Analysis that the simple linear regression is a suitable model to describe the following data(from Daniel,
3. Refer to Problem 11.(a) Construct the ANOVA table for the data in Problem 11.(b) Use the ANOVA table in (a) to evaluate the fitted model in Problem 11. Useα = 0.01.(c) Determine the observed level of significance (p-value) in (b).(d) Calculate the coefficient of determination R2. Give the
2. Refer to Problem 10.(a) Construct the ANOVA table for the data in Problem 10.(b) Use the ANOVA table in part (a) to evaluate the fitted model in Problem 10.Use α = 0.01.(c) Calculate the coefficient of determination R2.
1. Construct the ANOVA table for the data in Problem 12 and use results in this table to evaluate the fitted model in Problem 12. Use α = 0.05.
12. A study was made on the effect of temperature (X) on the yield (Y ) of a chemical process. The following data (coded) were collected (from Draper and Smith, 1981, used with permission):X: −5 −4 −3 −2 −1 0 1 2 3 4 5 Y: 1 5 4 7 10 8 9 13 14 13 18 15.3 Unbiased Estimator of σ2 637(a)
11. The weight (X) and total cholesterol level (Y) of 20 randomly selected females in the age group 30 and 40 are given below. Assume that the simple linear regression model is appropriate for these data.Weight, X: 135 140 128 143 150 155 147 146 137 144 Cholesterol, Y: 154 152 149 140 165 169 154
10. The purity (%) of oxygen produced by a fractional distillation process is believed to be related to percentage of hydrocarbons (%) in the main condenser of the processing unit. The data obtained on 20 samples are given below (from Introduction to Linear Regression Analysis by Montgomery et al.
9. Copper wire is the most widely used conductor since it has high conductivity and good mechanical properties. A wide range of cable applications require high tensile strength for copper wires. Mixing beryllium with copper increases the tensile 636 15 Simple Linear Regression Analysis strength of
8. In agriculture, it is important for obtaining a higher yield of certain crops that, at the time of planting, a proper distance be kept between plants. An agronomist conducted an experiment to investigate this for cotton crops. He divided a piece of land into small squares and sowed cotton seeds
7. The following data give the final scores for 12 randomly selected students in courses on probability and operations research (OR):Probability, X: 91 77 82 78 73 88 96 75 92 95 78 82 OR, Y: 86 75 86 76 75 89 87 91 83 90 84 94(a) Construct a scatter plot for these data. Does the scatter plot
6. In the paper manufacturing process, too much moisture left in the paper causes streaks that renders the paper unusable. Streaks can be avoided, for example, by slowing down the machine, over drying the paper, calibrating the dryer head, and so on. The following coded data give the amount of
5. The following data give the highest daytime temperature ◦F and amount of rainfall(in inches) on 10 randomly selected summer days at a tourist place in the northeastern United States:Temperature, X: 91 90 96 94 87 90 93 81 94 91 Rain, Y: 0.8 0.4 0.5 0.7 0.3 0.4 0.6 0.1 0.4 0.7 15.2 Fitting the
4. Recent studies show that high sound level (in decibels) makes humans prone to hypertension and heart attacks. For example, normal conversation level is 60 dB, for textile looms it is 105 dB, and for pneumatic chippers it is 115 dB. The following coded data give the noise level and the
3. A chemical engineer wants to investigate the relationship between the cooking time of paper pulp and the shear strength of the paper. She arranges to collect data for 12 batches of pulp that are cooked at the same temperature but for different periods of time. The data (coded) obtained is given
2. A chemist is interested in investigating the relationship between the reaction time and the yield of a chemical. He conducted 11 experiments with varying reaction time and measured the yield of the chemical. The data obtained are given below:Reaction time, X: 33 43 43 32 30 43 30 38 40 40 32
1. A team of physicians claims that a person’s excessive weight adversely affects his/her plasma glucose level. The data they obtained on 10 “overweight” persons are given below:Weight (lb), X: 181 180 189 188 229 192 223 231 212 225 Plasma glucose (mg/dL), Y: 206 162 181 199 214 146 210 165
15. A random sample of 12 students was selected from the freshman class at a liberal arts college. For each student, final scores in a first course in both calculus and physics are recorded and appear as given below. Do these data provide sufficient evidence to indicate that there is an association
14. The following data show the weight (in lb) and systolic blood pressure of 10 randomly selected male adults. Do the data provide sufficient evidence to indicate that there is a positive association between weight and systolic blood pressure? Use α=0.05.Weight (lb): 198 160 192 198 167 176 150
13. In a trial of two types of rain gauges, 65 of type A and 65 of type B were distributed at random over a certain region. In a given period, 14 storms occurred, and the average amounts of rain found in the two types of gauges were as shown below (from Brownlee, 1960). Test the hypothesis that the
12. Two analysts took repeated readings on the hardness of city water. The data are shown below. Determine whether one analyst has a tendency to read the instruments differently from the other, using the Mann–Whitney Wilcoxon test (data from Bowker and Lieberman, 1959). Use α = 0.05.Analyst A:
11. The 58 observations shown below were obtained by Millikan (1930) for the charge on an electron in 10−10 esu (CGS) units. By using runs above and below the median, test the hypothesis that the variation in this sequence is behaving randomly. Specify the value of α that you are using.4.781
10. During a 45-day production period in a cement plant, test cubes were taken each day, and the compressive strengths (in kg/cm2) of the test cubes were determined with the following results (data from Hald, 1952). Test the hypothesis of randomness at the 5% level of significance in this sequence
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