Questions and Answers of Regression Analysis

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
Consider the gasoline mileage data in Table B.3. Repeat Problem (parts $\mathrm{a}, \mathrm{b}$, and $\mathrm{c}$ ) using vehicle weight $x_{10}$ as the regressor variable. Based on a comparison of
Table B. 4 presents data for 27 houses sold in Erie, Pennsylvania.a. Fit a simple linear regression model relating selling price of the house to the current taxes $\left(x_{1}\right)$.b. Test for
The purity of oxygen produced by a fractional distillation process is thought to be related to the percentage of hydrocarbons in the main condensor of the processing unit. Twenty samples are shown
Consider the oxygen plant data in Problem and assume that purity and hydrocarbon percentage are jointly normally distributed random variables.a. What is the correlation between oxygen purity and
Consider the soft drink delivery time data in Table 2.10. After examining the original regression model (Example 2.9), one analyst claimed that the model was invalid because the intercept was not
The weight and systolic blood pressure of 26 randomly selected males in the age group 25-30 are shown below. Assume that weight and blood pressure (BP) are jointly normally distributed.a. Find a
The number of pounds of steam used per month at a plant is thought to be related to the average monthly ambient temperature. The past year's usages and temperatures follow.a. Fit a simple linear
Davidson ("Update on Ozone Trends in California's South Coast Air Basin," Air and Waste, 43, 226, 1993) studied the ozone levels in the South Coast Air Basin of California for the years 1976-1991. He
Hsuie, Ma, and Tsai ("Separation and Characterizations of Thermotropic Copolyesters of $p$-Hydroxybenzoic Acid, Sebacic Acid, and Hydroquinone," Journal of Applied Polymer Science, 56, 471-476, 1995)
Byers and Williams ("Viscosities of Binary and Ternary Mixtures of Polynomatic Hydrocarbons," Journal of Chemical and Engineering Data, 32, 349-354, 1987) studied the impact of temperature on the
Carroll and Spiegelman ("The Effects of Ignoring Small Measurement Errors in Precision Instrument Calibration," Journal of Quality Technology, 18, 170$173,1986)$ look at the relationship between the
Atkinson (Plots, Transformations, and Regression, Clarendon Press, Oxford, 1985) presents the following data on the boiling point of water $\left({ }^{\circ} \mathrm{F}\right)$ and barometric
On March 1, 1984, the Wall Street Journal published a survey of television advertisements conducted by Video Board Tests, Inc., a New York ad-testing company that interviewed 4000 adults. These
Table B. 17 Contains the Patient Satisfaction data used in Section 2.7.a. Fit a simple linear regression model relating satisfaction to age.b. Compare this model to the fit in Section 2.7 relating
Consider the fuel consumption data given in Table B.18. The automotive engineer believes that the initial boiling point of the fuel controls the fuel consumption. Perform a thorough analysis of these
Consider the wine quality of young red wines data in Table B.19. The winemakers believe that the sulfur content has a negative impact on the taste (thus, the overall quality) of the wine. Perform a
Consider the methanol oxidation data in Table B.20. The chemist believes that ratio of inlet oxygen to the inlet methanol controls the conversion process. Perform a through analysis of these data. Do
Consider the simple linear regression model $y=50+10 x+\varepsilon$ where $\varepsilon$ is NID $(0,16)$. Suppose that $n=20$ pairs of observations are used to fit this model. Generate 500 samples of
Consider the simple linear regression model $y=\beta_{0}+\beta_{1} x+\varepsilon$, with $E(\varepsilon)=0$, $\operatorname{Var}(\varepsilon)=\sigma^{2}$, and $\varepsilon$ uncorrelated.a. Show that
Consider the simple linear regression model $y=\beta_{0}+\beta_{1} x+\varepsilon$, with $E(\varepsilon)=0$, $\operatorname{Var}(\varepsilon)=\sigma^{2}$, and $\varepsilon$ uncorrelated.a. Show that
Suppose that we have fit the straight-line regression model $\hat{y}=\hat{\beta}_{0}+\hat{\beta}_{1} x_{1}$ but the response is affected by a second variable $x_{2}$ such that the true regression
Consider the maximum-likelihood estimator $\tilde{\sigma}^{2}$ of $\sigma^{2}$ in the simple linear regression model. We know that $\tilde{\sigma}^{2}$ is a biased estimator for $\sigma^{2}$.a. Show
Suppose that we are fitting a straight line and wish to make the standard error of the slope as small as possible. Suppose that the "region of interest" for $x$ is $-1 \leq x \leq 1$. Where should
Consider the data in Problem 2.12 and assume that steam usage and average temperature are jointly normally distributed.Data From Problem 2.12The number of pounds of steam used per month at a plant is
Prove that the maximum value of $R^{2}$ is less than 1 if the data contain repeated (different) observations on $y$ at the same value of $x$.
Consider the simple linear regression model \[y=\beta_{0}+\beta_{1} x+\varepsilon\] where the intercept $\beta_{0}$ is known.a. Find the least-squares estimator of $\beta_{1}$ for this model. Does
Consider the least-squares residuals $e_{i}=y_{i}-\hat{y}_{i}, i=1,2, \ldots, n$, from the simple linear regression model. Find the variance of the residuals $\operatorname{Var}\left(e_{i}\right)$.
Consider the baseball regression model from Section 2.8 and assume that wins and ERA are jointly normally distributed.a. Find the correlation between wins and team ERA.b. Test the hypothesis that
Consider the baseball data in Table B.22. Fit a regression model to team wins using total runs scored as the predictor. How does that model compare to the one developed in Section 2.8 using team ERA
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
Consider the rental price data in Table B.24. Assume that median home rental price and median price per square foot are jointly normally distributed.a. Find the correlation between home rental price
You have fit a linear regression model to a sample of 20 observations. The total sum of squares is 100 and the regression sum of squares is 80 . The estimate of the error variance isa. 1.5b. 1.2c.
You have fit a simple linear regression model to a sample of 25 observations. The value of the $t$-statistic for testing that the slope is zero is 2.75. An upper bound on the $P$-value for this test
A linear regression model with an intercept term will always pass through the centroid of the data.a. Trueb. False
The variance of the predicted response in a linear regression model is a minimum at the average value of the predictor variable.a. Trueb. False
The confidence interval on the mean response at a particular value of the predictor variable is always wider than the prediction interval on a new observation at the same point.a. Trueb. False
The method of least squares ensures that the estimators of the slope and intercept in a linear regression model are best linear unbiased estimator.a. Trueb. False
For any simple linear regression model that has an intercept, the sum of the residuals is always zero.a. Trueb. False
How does the use of a multiple regression model affect the number of degrees of freedom in a regression model? Contrast your response with that involving a simple regression model.
Provide an interpretation of a partial regression coefficient. What additional condition must be imposed that is not required in a simple regression model?
Explain the role the adjusted coefficient of determination plays in a multiple regression model. What value does it contribute to the interpretation of the model? What does it adjust for and why is
Define multicollinearity. Fully describe the problems it can cause and the effect each has on the model.
Discuss the various methods to detect multicollinearity. What advantages and disadvantages does each method of detection offer?
How can the ill-effects of multicollinearity be mitigated? Describe each potential remedy and note the advantages and disadvantages of each.
Define the Variance Inflation Factor. What function does it perform in regression analysis?
Describe fully how the Variance Inflation Factor for a variable can be determined.
How does the R2 value found in a VIF test differ from that for the entire model?
What is meant by Analysis of Variance? What is its purpose? What exactly is the hypothesis in an ANOVA test? What does it mean if the null is not rejected? How do you interpret the rejection of the
How is the F-value in an ANOVA calculated? Why does a large Fvalue result in a rejection of the null? Use a graph in your response.

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