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regression analysis
Questions and Answers of
Regression Analysis
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 $\operatorname{Var}(\hat{\mathbf{y}})=\sigma^{2} \mathbf{H}$.
Prove that the matrices $\mathbf{H}$ and $\mathbf{I}-\mathbf{H}$ are idempotent, that is, $\mathbf{H H}=\mathbf{H}$ and $(\mathbf{I}-\mathbf{H})(\mathbf{I}-\mathbf{H})=\mathbf{I}-\mathbf{H}$.
For the simple linear regression model, show that the elements of the hat matrix are\[h_{i j}=\frac{1}{n}+\frac{\left(x_{i}-\bar{x}\right)\left(x_{j}-\bar{x}\right)}{S_{x x}} \text { and } h_{i
Consider the multiple linear regression model $\mathbf{y}=\mathbf{X} \boldsymbol{\beta}+\boldsymbol{\varepsilon}$. Show that the least-squares estimator can be written
Show that the residuals from a linear regression model can be expressed as $\mathbf{e}=(\mathbf{I}-\mathbf{H}) \boldsymbol{\varepsilon}$.
For the multiple linear regression model, show that $S S_{\mathrm{R}}(\boldsymbol{\beta})=\mathbf{y}^{\prime} \mathbf{H y}$.
Prove that $R^{2}$ is the square of the correlation between $\mathbf{y}$ and $\hat{\mathbf{y}}$.
Constrained least squares. Suppose we wish to find the least-squares estimator of $\boldsymbol{\beta}$ in the model $\mathbf{y}=\mathbf{X} \boldsymbol{\beta}+\boldsymbol{\varepsilon}$ subject to a
Let $\mathbf{x}_{j}$ be the $j$ th row of $\mathbf{X}$, and $\mathbf{X}_{-j}$ be the $\mathbf{X}$ matrix with the $j$ th row removed. Show
Consider the following two models where $E(\boldsymbol{\varepsilon})=\mathbf{0}$ and $\operatorname{Var}(\boldsymbol{\varepsilon})=\sigma^{2} \mathbf{I}$ :Model A: $\mathbf{y}=\mathbf{X}_{1}
Suppose we fit the model $\mathbf{y}=\mathbf{X}_{1} \boldsymbol{\beta}_{2}+\boldsymbol{\varepsilon}$ when the true model is actually given by $\mathbf{y}=\mathbf{X}_{1}
Consider a correctly specified regression model with $p$ terms, including the intercept. Make the usual assumptions about $\varepsilon$. Prove that\[\sum_{i=1}^{n}
Let $R_{j}^{2}$ be the coefficient of determination when we regress the $j$ th regressor on the other $k-1$ regressors. Show that the $j$ th variance inflation factor may be expressed
Consider the hypotheses for the general linear model, which are of the form\[H_{0}: \mathbf{T} \beta=\mathbf{c}, \quad H_{1}: \mathbf{T} \beta eq \mathbf{c}\]where $\mathbf{T}$ is a $q \times p$
Consider the 2016 major league baseball data in Table B.22. While team ERA was useful in predicting the number of games that a team wins, there are some other measures of team performance, including
Table B. 24 contains data on median family home rental price and other data for 51 US cities. Fit a linear regression model using the median home rental price as the response variable and median
You have fit a linear regression model with three predictors to a sample of 50 observations. The total sum of squares is 150 and the regression sum of squares is 120 . The estimate of the error
Consider the regression model in Problem 3.43. The value of the adjusted $R^{2}$ isData From Problem 3.43You have fit a linear regression model with three predictors to a sample of 50 observations.
Consider the simple regression model fit to the National Football League team performance data in Problem 2.1.Data From Problem 2.1Table B. 1 gives data concerning the performance of the 26 National
Consider the multiple regression model fit to the National Football League team performance data in Problem 3.1. Problem 3.1Consider the National Football League data in Table B.1.a. Construct a
Consider the simple linear regression model fit to the solar energy data in Problem 2.3. Problem 2.3Table B. 2 presents data collected during a solar energy project at Georgia Tech.a. Construct a
Consider the multiple regression model fit to the gasoline mileage data in Problem 3.5.Problem 3.5 Consider the gasoline mileage data in Table B.3.a. Construct a normal probability plot of the
Consider the multiple regression model fit to the house price data in Problem 3.7.Problem 3.7Consider the house price data in Table B.4.a. Construct a normal probability plot of the residuals. Does
Consider the simple linear regression model fit to the oxygen purity data in Problem 2.7.Problem 2.7The purity of oxygen produced by a fractional distillation process is thought to be related to the
Consider the simple linear regression model fit to the weight and blood pressure data in Problem 2.10.Problem 2.10The weight and systolic blood pressure of 26 randomly selected males in the age group
Consider the simple linear regression model fit to the steam plant data in Problem 2.12.Problem 2.12The number of pounds of steam used per month at a plant is thought to be related to the average
Consider the simple linear regression model fit to the ozone data in Problem 2.13.Problem 2.13Davidson ("Update on Ozone Trends in California's South Coast Air Basin," Air and Waste, 43, 226, 1993)
Consider the simple linear regression model fit to the copolyester viscosity data in Problem 2.14.Problem 2.14Hsuie, Ma, and Tsai ("Separation and Characterizations of Thermotropic Copolyesters of
Consider the simple linear regression model fit to the toluene-tetralin viscosity data in Problem 2.15.Problem 2.15Byers and Williams ("Viscosities of Binary and Ternary Mixtures of Polynomatic
Consider the simple linear regression model fit to the tank pressure and volume data in Problem 2.16.Problem 2.16Carroll and Spiegelman ("The Effects of Ignoring Small Measurement Errors in Precision
Problem 3.8 asked you to fit two different models to the chemical process data in Table B.5. Perform appropriate residual analyses for both models. Discuss the results of these analyses. Calculate
Coteron, Sanchez, Martinez, and Aracil ("Optimization of the Synthesis of an Analogue of Jojoba Oil Using a Fully Central Composite Design," Canadian Journal of Chemical Engineering, 1993) studied
Derringer and Suich ("Simultaneous Optimization of Several Response Variables," Journal of Quality Technology, 1980) studied the relationship of an abrasion index for a tire tread compound in terms
Myers, Montgomery and Anderson-Cook (Response Surface Methodology 4th edition, Wiley, New York, 2016) discuss an experiment to determine the influence of five factors:$x_{1}$ - acid bath
Consider the test for lack of fit. Find $E\left(M S_{\mathrm{PE}}\right)$ and $E\left(M S_{\mathrm{LOF}}\right)$.
Table B. 14 contains data on the transient points of an electronic inverter. Using only the regressors $x_{1}, \ldots, x_{4}$, fit a multiple regression model to these data.a. Investigate the
Consider the air pollution and mortality data given in Problem 3.15 and Table B. 15 .Problem 3.15McDonald and Ayers [1978] present data from an early study that examined the possible link between air
Consider the life expectancy data given in Problem 3.16 and Table B.16.Problem 3.16Rossman [1994] presents an interesting study of average life expectancy of 40 countries. Table B. 16 gives the data.
Consider the fuel consumption data in Table B.18. For the purposes of this exercise, ignore regressor $x_{1}$. Perform a thorough residual analysis of these data. What conclusions do you draw from
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 residual analysis of these data. What conclusions do
Consider the methanol oxidation data in Table B.20. Perform a thorough analysis of these data. What conclusions do you draw from this residual analysis? x x2 X3 X4 0 454 8.8 3.90 1.30 1.1 0 474 8.2
Consider the regression model fit to the baseball data in Table B.22, using team ERA to predict the number of wins.a. Construct a normal probability plot of the residuals. Is there any indication of
Consider the multiple linear regression model fit to the baseball data in Problem 3.41.Problem 3.41Consider the 2016 major league baseball data in Table B.22. While team ERA was useful in predicting
Consider the simple linear regression model fit to the rental price data from Problem 2.36.Data From Problem 2.36Table B.24 contains data on median family home rental price and other data for 51 US
Consider the multiple linear regression model fit to the rental price data in Problem 3.42.Problem 3.42 Table B.24 contains data on median family home rental price and other data for 51 US cities.
Consider the simple linear regression model for the baseball data using Team ERA as the predictor. Find the value of the PRESS statistic and the $R^{2}$ based on PRESS for this model. What
Consider the multiple linear regression model fit to the baseball data in Problem 3.41.Problem 3.41Consider the 2016 major league baseball data in Table B.22. While team ERA0 was useful in predicting
Consider the multiple linear regression model for the rental price data in Problem 3.42.Problem 3.42 Table B.24 contains data on median family home rental price and other data for 51 US cities. Fit
Table B. 1 gives data concerning the performance of the 26 National Football League teams in 1976. It is suspected that the number of yards gained rushing by opponents $\left(x_{8}\right)$ has an
Suppose we would like to use the model developed in Problem 2.1 to predict the number of games a team will win if it can limit opponents' yards rushing to 1800 yards. Find a point estimate of the
Table B. 2 presents data collected during a solar energy project at Georgia Tech.a. Fit a simple linear regression model relating total heat flux $y$ (kilowatts) to the radial deflection of the
Table B. 3 presents data on the gasoline mileage performance of 32 different automobiles.a. Fit a simple linear regression model relating gasoline mileage $y$ (miles per gallon) to engine
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