Questions and Answers of Regression Analysis

Use the all-possible-regressions method to select a subset regression model for the Belle Ayr liquefaction data in Table B.5. Evaluate the subset models using the $C_{p}$ criterion. Justify your
Analyze the tube-flow reactor data in Table B. 6 using all possible regressions. Evaluate the subset models using the $R_{p}^{2}, C_{p}$, and $M S_{\text {Res }}$ criteria. Justify your choice of
Analyze the air pollution and mortality data in Table B. 15 using all possible regressions. Evaluate the subset models using the $R_{p}^{2}, C_{p}$, and $M S_{\text {Res }}$ criteria. Justify
Consider the all-possible-regressions analysis of Hald's cement data in Example 10.1. If the objective is to develop a model to predict new observations, which equation would you recommend and
Consider the all-possible-regressions analysis of the National Football League data in Problem 10.2. Identify the subset regression models that are $R^{2}$ adequate (0.05).Data From Problem
Suppose that the full model is $y_{i}=\beta_{0}+\beta_{1} x_{i 1}+\beta_{2} x_{i 2}+\varepsilon_{i}, i=1,2, \ldots, n$, where $x_{i 1}$ and $x_{i 2}$ have been coded so that
Table B. 11 presents data on the quality of Pinot Noir wine.a. Build an appropriate regression model for quality $y$ using the all-possibleregressions approach. Use $C_{p}$ as the model selection
Use the wine quality data in Table B. 11 to construct a regression model for quality using the stepwise regression approach. Compare this model to the one you found in Problem 10.4, part a.Data From
Rework Problem 10.14, part a, but exclude the region information.a. Comment on the difference in the models you have found. Is there indication that the region information substantially improves the
Table B. 12 presents data on a heat treating process used to carburize gears. The thickness of the carburized layer is a critical factor in overall reliability of this component. The response
Table B. 13 presents data on the thrust of a jet turbine engine and six candidate regressors. Use all possible regressions and the $C_{p}$ criterion to find an appropriate regression model for
Table B. 14 presents data on the transient points of an electronic inverter. Use all possible regressions and the $C_{p}$ criterion to find an appropriate regression model for these data.
Consider the soft drink delivery time data in Example 3.1.Example 3.1a. Find the simple correlation between cases $\left(x_{1}\right)$ an distance $\left(x_{2}\right)$.b. Find the variance
Consider the Hald cement data in Table B.21.a. From the matrix of correlations between the regressors, would you suspect that multicollinearity is present?b. Calculate the variance inflation
Using the Hald cement data (Example 10.1), find the eigenvector associated with the smallest eigenvalue of $\mathbf{X}^{\prime} \mathbf{X}$. Interpret the elements of this vector. What can you say
Find the condition indices and the variance decomposition proportions for the Hald cement data (Table B.21), assuming centered regressors. What can you say about multicollinearity in these data?
Repeat Problem 9.4 without centering the regressors and compare the results. Which approach do you think is better?Data From Problem 9.4Find the condition indices and the variance decomposition
Use the regressors $x_{2}$ (passing yardage), $x_{7}$ (percentage of rushing plays), and $x_{8}$ (opponents' yards rushing) for the National Football League data in Table B.1.a. Does the
Consider the gasoline mileage data in Table B.3.a. Does the correlation matrix give any indication of multicollinearity?b. Calculate the variance inflation factors and the condition number of
Using the gasoline mileage data in Table B. 3 find the eigenvectors associated with the smallest eigenvalues of $\mathbf{X}^{\prime} \mathbf{X}$. Interpret the elements of these vectors. What can
Use the gasoline mileage data in Table B. 3 and compute the condition indices and variance-decomposition proportions, with the regressors centered. What statements can you make about
Analyze the housing price data in Table B. 4 for multicollinearity. Use the variance inflation factors and the condition number of $\mathbf{X}^{\prime} \mathbf{X}$. y X1 X2 X3 X4 X5 X6 X7 Xg 25.9
Analyze the chemical process data in Table B. 5 for evidence of multicollinearity. Use the variance inflation factors and the condition number of $\mathbf{X}^{\prime} \mathbf{X}$. Run No. y X x2 X3
Analyze the patient satisfaction data in Table B. 17 for multicollinearity. Satisfaction Age Severity Surgical-Medical Anxiety 68 55 50 0 2.1 77 46 24 1 2.8 96 30 46 1 3.3 80 35 48 1 4.5 43 59 58 0 2
Analyze the fuel consumption data in Table B. 18 for multicollinearity. y X2 X3 X4 xs X6 X7 Xg 343 0 52.8 811.7 2.11 220 261 87 1.8 356 1 52.8 811.7 2.11 220 261 87 1.8 344 0 50.0 821.3 2.11 223 260
Analyze the wine quality of young red wines data in Table B. 19 for multicollinearity. y X2 X3 X4 x6 X7 Xg Xg x10 19.2 0 3.85 66 9.35 5.65 2.40 3.25 0.33 19 0.065 18.3 0 3.73 79 11.15 6.95 3.15 3.80
Analyze the methanol oxidation data in Table B. 20 for multicollinearity. x1 X2 X3 X4 xs 0 454 8.8 3.90 1.30 1.1 474 8.2 3.68 1.16 4.2 524 7.0 2.78 1.25 94.2 503 7.4 2.27 1.57 20.7 493 7.6 2.40 1.55
The table below shows the condition indices and variance decomposition proportions for the acetylene data using centered regressors. Use this information to diagnose multicollinearity in the data and
Apply ridge regression to the Hald cement data in Table B.21.a. Use the ridge trace to select an appropriate value of $k$. Is the final model a good one?b. How much inflation in the residual sum of
Use ridge regression on the Hald cement data (Table B.21) using the value of $k$ in Eq. (9.8). Compare this value of $k$ value selected by the ridge trace in Problem 9.17. Does the final model
Estimate the parameters in a model for the gasoline mileage data in Table B. 3 using ridge regression.a. Use the ridge trace to select an appropriate value of $k$. Is the resulting model
Estimate the parameters in a model for the gasoline mileage data in Table B. 3 using ridge regression with the value of $k$ determined by Eq. (9.8). Does this model differ dramatically from the one
Estimate model parameters for the Hald cement data (Table B.21) using principal-component regression.a. What is the loss in $R^{2}$ for this model compared to least squares?b. How much shrinkage in
Estimate the model parameters for the gasoline mileage data using principalcomponent regression.a. How much has the residual sum of squares increased compared to least squares?b. How much shrinkage
Consider the air pollution and mortality data given in Table B.15.a. Is there a problem with collinearity? Discuss how you arrived at this decision.b. Perform a ridge trace on these data.c. Select a
Consider the air pollution and mortality data given in Table B.15.a. Is there a problem with collinearity? Discuss how you arrived at this decision.b. Perform a ridge trace on these data.c. Select a
The pure shrinkage estimator is defined as $\hat{\beta}_{s}=c \hat{\beta}$, were $0 \leq c \leq 1$ is a constant chosen by the analyst. Describe the kind of shrinkage that this estimator
Show that the pure shrinkage estimator (Problem 9.25) is the solution toData From Problem 9.25The pure shrinkage estimator is defined as $\hat{\beta}_{s}=c \hat{\beta}$, were $0 \leq c \leq 1$ is
The mean square error criterion for ridge regression is\[E\left(L_{1}^{2}\right)=\sum_{j=1}^{p} \frac{\lambda_{j}}{\left(\lambda_{j}+k\right)^{2}}+\sum_{j=1}^{p} \frac{\alpha_{j}^{2}
Consider the mean square error criterion for generalized ridge regression. Show that the mean square error is minimized by choosing $k_{j}=\sigma^{2} / \alpha_{j}^{2}, j=1$, $2, \ldots, p$.
Show that if $\mathbf{X}^{\prime} \mathbf{X}$ is in correlation form, $\boldsymbol{\Lambda}$ is the diagonal matrix of eigenvalues of $\mathbf{X}^{\prime} \mathbf{X}$, and $\mathbf{T}$ is the
Formally show that\[D_{i}=\frac{r_{i}}{p} \frac{h_{i i}}{1-h_{i i}}\]
Table B. 14 contains data concerning the transient points of an electronic inverter. Fit a regression model to all 25 observations but only use $x_{1}-x_{4}$ as the regressors. Investigate this
Perform a thorough influential analysis of the air pollution and mortality data given in Table B.15. Perform any appropriate transformations. Discuss your results. City Mort Precip Educ Nonwhite Nox
Consider the patient satisfaction data in Table B.17. Fit a regression model to the satisfaction response using age and severity as the predictors. Perform an influence analysis of the date and
Chemical and mechanical engineers often need to know the vapor pressure of water at various temperatures (the "infamous" steam tables can be used for this). Below are data on the vapor pressure of
An article in the Journal of Pharmaceutical Sciences $(\mathbf{8 0}, 971-977,1991)$ presents data on the observed mole fraction solubility of a solute at a constant temperature, along with
Pet-Pro is a company that imports and markets pet food feeders to customers in Malaysia. The food feeder, which is called "Smart Feeder," allows pet owners to feed pre-determined quantities of food
Consider the regression model (8.8) described in Example 8.3 Graph the response function for this model and indicate the role the model parameters play in determining the shape of this
Consider the regression models described in Example 8.4.Example 8.4a. Graph the response function associated with Eq. (8.10).Equation (8.10)b. Graph the response function associated with Eq.
Consider the delivery time data in Example 3.1. In Section 4.2.5 noted that these observations were collected in four cities, San Diego, Boston, Austin, and Minneapolis.Example 3.1 a. Develop a
Consider the automobile gasoline mileage data in Table B.3.a. Build a linear regression model relating gasoline mileage $y$ to engine displacement $x_{1}$ and the type of transmission $x_{11}$.
Consider the automobile gasoline mileage data in Table B.3.a. Build a linear regression model relating gasoline mileage $y$ to vehicle weight $x_{10}$ and the type of transmission $x_{11}$. Does the
Consider the National Football League data in Table B.1. Build a linear regression model relating the number of games won to the yards gained rushing by opponents $x_{8}$, the percentage of rushing
Piecewise Linear Regression. In Example 7.3 we showed how a linear regression model with a change in slope at some point $t\left(x_{\min }Example 7.3 An important special case of practical interest
Continuation of Problem 8.7 . Show how indicator variables can be used to develop a piecewise linear regression model with a discontinuity at the join point $t$.Problem 8.7Piecewise Linear
Suppose that a one-way analysis of variance involves four treatments but that a different number of observations (e.g., $n_{i}$ ) has been taken under each treatment. Assuming that $n_{1}=3, n_{2}=2,
Alternate Coding Schemes for the Regression Approach to Analysis of Variance. Consider Eq. (8.18), which represents the regression model corresponding to an analysis of variance with three treatments
Montgomery [2020] presents an experiment concerning the tensile strength of synthetic fiber used to make cloth for men's shirts: The strength is thought to be affected by the percentage of cotton in
Two-Way Analysis of Variance. Suppose that two different sets of treatments are of interest. Let $y_{i j k}$ be the $k$ th observation level $i$ of the first treatment type and level $j$ of
Table B. 11 presents data on the quality of Pinot Noir wine.a. Build a regression model relating quality $y$ to flavor $x_{4}$ that incorporates the region information given in the last column.
Using the wine quality data from Table B.11, fit a model relating wine quality $y$ to flavor $x_{4}$ using region as an allocated code, taking on the values shown in the table $(1,2,3)$. Discuss the
Consider the life expectancy data given in Table B.16. Create an indicator variable for gender. Perform a thorough analysis of the overall average life expectancy. Discuss the results of this
Smith et al. [1992] discuss a study of the ozone layer over the Antarctic. These scientists developed a measure of the degree to which oceanic phytoplankton production is inhibited by exposure to
Table B. 17 contains hospital patient satisfaction data. Fit an appropriate regression model to the satisfaction response using age and severity as the regressors and account for the medical versus
Consider the fuel consumption data in Table B.18. Regressor $x_{1}$ is an indicator variable. Perform a thorough analysis of these data. What conclusions do you draw from this analysis? y X2 X3 X4
Consider the wine quality of young red wines data in Table B.19. Regressor $x_{1}$ is an indicator variable. Perform a thorough analysis of these data. What conclusions do you draw from this
Consider the methanol oxidation data in Table B.20. Perform a thorough analysis of these data. What conclusions do you draw from this analysis? x x2 3 X4 Xs y 0 454 8.8 3.90 1.30 1.1 0 474 8.2 3.68
Table B.23 contains player efficiency ratings (PER) from the 2016-17 and 2017-18 NBA combine that evaluates 60 rookies hoping to be drafted by NBA teams. PER is a measure of a player's per-minute
Use the NBA PER data introduced in Problem 8.21 and consider the model found in part $\mathrm{c}$ of that problem. There are some potential outliers in the data (the first observation is an obvious
Use the NBA PER data introduced in Problem 8.21 and consider the model found in Problem 8.22. After the outliers are removed it is not obvious that all of the terms in the model are important. Refine
The following table gives the vapor pressure of water for various temperaturesa. Plot a scatter diagram. Does it seem likely that a straight-line model will be adequate? b. Fit the straight-line
Consider the three modelsa. $y=\beta_{0}+\beta_{1}(1 / x)+\varepsilon$b. $1 / y=\beta_{0}+\beta_{1} x+\varepsilon$c. $y=x /\left(\beta_{0}-\beta_{1} x\right)+\varepsilon$All of these models can
How are databases useful for managers?
Consider the kinematic viscosity data in Table B.10.a. Perform a thorough residual analysis of these data.b. Identify the most appropriate transformation for these data. Fit this model and repeat the
French and Schultz ("Water Use Efficiency of Wheat in a Mediterranean-type Environment, I The Relation between Yield, Water Use, and Climate," Australian Journal of Agricultural Research, 35, 743-64)
Consider the National Football League data in Table B.1.a. Fit a multiple linear regression model relating the number of games won to the team's passing yardage $\left(x_{2}\right)$, the percentage
Using the results of Problem 3.1, show numerically that the square of the simple correlation coefficient between the observed values $y_{i}$ and the fitted values $\hat{y}_{i}$ equals $R^{2}$.Data
Refer to Problem 3.1.Data From Problem 3.1Consider the National Football League data in Table B.1.a. Find a $95 % \mathrm{CI}$ on $\beta_{7}$.b. Find a $95 %$ CI on the mean number of games won by a
Reconsider the National Football League data from Problem 3.1. Fit a model to these data using only $x_{7}$ and $x_{8}$ as the regressors.Data From Problem 3.1Consider the National Football League
Consider the gasoline mileage data in Table B.3.a. Fit a multiple linear regression model relatmg gasoline mileage $y$ (miles per gallon) to engine displacement $x_{1}$ and the number of carburetor
In Problem 2.4 you were asked to compute a $95 %$ CI on mean gasoline prediction interval on mileage when the engine displacement $x_{1}=275$ in. $^{3}$ Compare the lengths of these intervals to the
Consider the house price data in Table B.4.a. Fit a multiple regression model relating selling price to all nine regressors.b. Test for significance of regression. What conclusions can you draw?c.
The data in Table B. 5 present the performance of a chemical process as a function of several controllable process variables.a. Fit a multiple regression model relating $\mathrm{CO}_{2}$ product
The concentration of $\mathrm{NbOCl}_{3}$ in a tube-flow reactor as a function of several controllable variables is shown in Table B.6.a. Fit a multiple regression model relating concentration of
The quality of Pinot Noir wine is thought to be related to the properties of clarity, aroma, body, flavor, and oakiness. Data for 38 wines are given in Table B. 11 .a. Fit a multiple linear
An engineer performed an experiment to determine the effect of $\mathrm{CO}_{2}$ pressure, $\mathrm{CO}_{2}$ temperature, peanut moisture, $\mathrm{CO}_{2}$ flow rate, and peanut particle size on the
A chemical engineer studied the effect of the amount of surfactant and time on clathrate formation. Clathrates are used as cool storage media. Table B. 8 summarizes the experimental results.a. Fit a
An engineer studied the effect of four variables on a dimensionless factor used to describe pressure drops in a screen-plate bubble column. Table B. 9 summarizes the experimental results.a. Fit a
The kinematic viscosity of a certain solvent system depends on the ratio of the two solvents and the temperature. Table B. 10 summarizes a set of experimental results.a. Fit a multiple linear
McDonald and Ayers [1978] present data from an early study that examined the possible link between air pollution and mortality. Table B. 15 summarizes the data. The response MORT is the total
Rossman [1994] presents an interesting study of average life expectancy of 40 countries. Table B. 16 gives the data. The study has three responses: LifeExp is the overall average life expectancy.
Consider the patient satisfaction data in Table B.17. For the purposes of this exercise, ignore the regressor "Medical-Surgical." Perform a thorough analysis of these data. Please discuss any
Consider the fuel consumption data in Table B.18. For the purposes of this exercise, ignore regressor $x_{1}$. Perform a thorough analysis of these data. What conclusions do you draw from this
Consider the wine quality of young red wines data in Table B.19. For the purposes of this exercise, ignore regressor $x_{1}$. Perform a thorough analysis of these data. What conclusions do you draw
Consider the methanol oxidation data in Table B.20. Perform a thorough analysis of these data. What conclusions do you draw from this analysis? x x2 X3 X4 y 0 454 8.8 3.90 1.30 1.1 0 474 8.2 3.68
Show that an alternate computing formula for the regression sum of squares in a linear regression model is \[S S_{\mathrm{R}}=\sum_{i=1}^{n} \hat{y}_{i}^{2}-n \bar{y}^{2}\]
Consider the multiple linear regression model \[ y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{3}+\beta_{4} x_{4}+\varepsilon \] Using the procedure for testing a general linear
Suppose that we have two independent samples, sayTwo models can be fit to these samples,\[\begin{gathered}y_{i}=\beta_{0}+\beta_{1} x_{i}+\varepsilon_{i}, \quad i=1,2, \ldots, n_{2}
Show that $\operatorname{Var}(\hat{\mathbf{y}})=\sigma^{2} \mathbf{H}$.

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