christian-mann · April 18, 2012 06:57
diff --git a/hw9.tex b/hw9.tex

 \documentclass{exam}
 \usepackage[margin=1in]{geometry}
 \usepackage{amsmath}
 \usepackage{booktabs}
 \usepackage{listings}
 \usepackage{graphicx}
 \usepackage{paralist}
 \title{Statistics Homework 9}
 \author{Christian Mann}
 \date{\today}
 \begin{document}
 	\maketitle
 	\printanswers
 	\begin{questions}
 		\question \textbf{Question 12-36.} Mist (airborne droplets or aerosols) is generated when metal-removing fluids are used in machining operations to cool and lubricate the tool and workpiece. %heh, heh
 		Mist generation is a concern to OSHA, which has recently lowered substantially the workplace standard. The article ``Variables Affecting Mist Generation from Metal Removal Fluids'' (\textit{Lubrication Engr.}, 2002: 10-17) gave the accompanying data on $x$ = fluid-flow velocity for a 5\% soluble oil (cm/sec) and $y$ = the extent of mist droplets having diameters smaller than 10$\mu$m (mg/m$^3$):
 		\begin{center}
 		\begin{tabular}{c | c c c c c c c}
 			\toprule
 			$x$ & 89 & 177 & 189 & 354 & 362 & 442 & 965 \\
 			\midrule
 			$y$ & .40 & .60 & .48 & .66 & .61 & .69 & .99 \\
 			\bottomrule
 		\end{tabular}
 		\end{center}
 		\begin{parts}
 			\part The investigators performed a simple linear regression analysis to relate the two variables. Does a scatter plot of the data support this strategy?
 			\part What proportion of observed variation in mist can be attributed to the simple linear regression relationshipt between velocity and mist?
 			\part The investigators were particularly interested in the impact on mist of increasing velocity from 100 to 1000 (a factor of 10 corresponding to difference between the smallest and largest $x$ values in the sample). When $x$ increases in this way, is there substantial evidence that the true average increase in $y$ is less than .6?
 			\part Estimate the true average change in mist associated with a 1 cm/sec increase in velocity, and do so in a way that conveys information about precision and reliability.
 		\end{parts}
 	\question \textbf{Question 12-41.} \begin{parts}
 		\part Verify that $E(\hat{\beta_1}) = \beta_1$ by using the rules of expected value from Chapter 5.
 		\part Use the rules of variance from Chapter 5 to verify the expression for $V(\hat{\beta_1})$ given in this section.
 		\end{parts}
 	\question \textbf{Question 12-50.} Silicon-germanium alloys have been used in certain types of solar cells. The paper ``Silicon-Germanium Films Deposited by Low-Frequency Plasma-Enhanced Chemical Vapor Deposition'' (\textit{J. of Material Res.}, 2006: 88--104) reported on a study of various structural and electrical properties. Consider the accompanying data on $x$ = Ge concentration in solid phase (ranging from 0 to 1) and $y$ = Fermi level position (eV):
 		\begin{center}
 		\begin{tabular}{c|c c c c c c c c c c c c c c}
 			$x$ & 0 & .42 & .23 & .33 & .62 & .60 & .45 & .87 & .90 & .79 & 1 & 1 & 1 \\
 			$y$ & .62 & .53 & .61 & .59 & .50 & .55 & .59 & .31 & .43 & .46 & .23 & .22 & .19 \\
 		\end{tabular}
 		\end{center}
 		A scatter plot shows a substantial linear relationship. Here is Minitab output from a least squares fit. [\textit{Note:} There are several inconsistencies between the data given in the paper, the plot that appears there, and the summary information about a regression analysis.]
 		\begin{verbatim}
 The regression equation is
 Fermi pos = 0.7217 - 0.4327 Ge conc

 S = 0.0737573 R-Sq = 80.2% R-Sq(adj) = 78.4%

 Analysis of Variance

 Source      DF        SS        MS	    F	     P
 Regression   1  0.241728  0.241728  44.43  0.000
 Error       11  0.059842  0.005440
 Total       12  0.301569
 		\end{verbatim}
 		\begin{parts}
 			\part Obtain an interval estimate of the expected change in Fermi-level position associated with an increase of .1 in Ge concentration, and interpret your result.
 			\part Obtain an interval estimate for mean Fermi-level position when concentration is .50, and interpret your estimate.
 			\part Obtain an interval of plausible values for position resulting from a single observation to be made when concentration is .50, interpret your interval, and compare to the interval of (b).
 			\part Obtain simultaneous CIs for expected position when concentration is .3, .5, and .7; the joint confidence level should be at least 97\%.
 		\end{parts}
 	\question Plasma etching is essential to the fine-line pattern transfer in current semiconductor processes. The article ``Ion Beam-Assisted Etching of Aluminum with Chlorine'' (\textit{J. of the Electrochem. Soc.}, 1985: 2010--2012) gives the accompanying data (read from a graph) on chlorine flow ($x$, in SCCM) through a nozzle used in the eching mechanism and etch rate ($y$, in 100 A/min).
 		\begin{center}
 		\begin{tabular}{c|c c c c c c c c c}
 			\toprule
 			$x$ & 1.5 & 1.5 & 2.0 & 2.5 & 2.5 & 3.0 & 3.5 & 3.5 & 4.0 \\
 			\midrule
 			$y$ & 23.0 & 24.5 & 25.0 & 30.0 & 33.5 & 40.0 & 40.5 & 47.0 & 49.0 \\
 			\bottomrule
 		\end{tabular}
 		\end{center}
 		The summary statistics are $\Sigma x_i = 24.0, \Sigma y_i = 312.5, \Sigma x_i^2 = 70.50, \Sigma x_iy_i = 902.5, \Sigma y_i^2 = 11,626.75, \hat{\beta_0} = 6.448718, \hat{\beta_1} = 10.602564$. \begin{parts}
 			\part Does the simple linear regression model specify a useful relationshipt between chlorine flow and etch rate?
 			\part Estimate the true average change in etch rate associated with a 1-SCCM increase in flow rate using a 95\% confidence interval, and interpret the interval.
 			\part Calculate a 95\% CI for $\mu_{Y \cdot 3.0}$, the true average etch rate when flow = 3.0. Has this average been precisely estimated?
 			\part Calculate a 95\% PI for a single future observation on etch rate to be made when flow = 3.0. Is the prediction likely to be accurate?
 			\part Would the 95\% CI and PI when flow = 2.5 be wider or narrower than the corresponding intervals of parts (c) and (d)? Answer without actually computing the intervals.
 			\part Would you recommend calculating a 95\% PI for a flow of 6.0? Explain.
 		\end{parts}
 	\question \textbf{Question 12-62.} Hydrogen content is conjectured to be an important factor in porosity of aluminum alloy castings. The article ``The Reduced Pressure Test as a Measuring Tool in the Evaluation of Porosity/Hydrogen Content in A1-7 Wt Pct Si-10 Vol Pct SiC(p) Metal Matrix Composite'' (\textit{Metallurgical Trans.}, 1993: 1857-1868) gives the accompanying data on $x$ = content and $y$ = gas porosity for one particular measurement technique.
 		\begin{center}
 		\begin{tabular}{c|c c c c c c c c c c c c c c}
 			\toprule
 			$x$ & .18 & .20 & .21 & .21 & .21 & .22 & .23 & .23 & .24 & .24 & .25 & .28 & .30 & .37 \\
 			\midrule
 			$y$ & .46 & .70 & .41 & .45 & .55 & .44 & .24 & .47 & .22 & .80 & .88 & .70 & .72 & .75 \\
 			\bottomrule
 		\end{tabular}
 		\end{center}
 		Minitab gives the following output in response to a Correlation command:
 		\begin{verbatim}
 Correlation of Hydrocon and Porosity = 0.449
 		\end{verbatim}
 		\begin{parts}
 			\part Test at level .05 to see whether the population correlation coefficient differs from 0.
 			\part If a simple linear regression analysis had been carried out, whate percentage of observed variation in porosity could be attributed to the model relationship?
 		\end{parts}
 	\question \textbf{Question 12-72.} The SAS output at the bottom of this page is based on data from the article ``Evidence for and the Rate of Denitrification in the Arabian Sea'' (\textit{Deep Sea Research}, 1978: 431-435). The variables under study are $x$ = salinity level (\%) and $y$ = nitrate level ($\mu$M/L). 
 		\begin{parts}
 			\part What is the sample size $n$? [\textit{Hint}: Look for degrees of freedom for SSE.]
 			\part Calculate a point estimate of expected nitrate level when salinity level is 35.5.
 			\part Does there appear to be a useful linear relationship between the two variables?
 			\part What is the value of the sample correlation coefficient?
 			\part Would you use the simple linear regression model to draw conclusions when the salinity level is 40?
 		\end{parts}
 	\question \textbf{Question 13-20.} Exercise 14 presented data on body weight $x$ and metabolic clearance rate/body weight $y$. Consider the following intrinsically linear functions for specifying the relationship between the two variables: \begin{inparaenum}[(a)] 
 		\item ln($y$) versus $x$, 
 		\item ln($y$) versus ln($x$), 
 		\item $y$ versus ln($x$), 
 		\item $y$ versus 1/$x$, and 
 		\item ln($y$) versus 1/$x$ 
 		\end{inparaenum}. Use any appropriate diagnostic plots and analyses to decide which of these functions you would select to specify a probabilistic model. Explain your reasoning.
 	\question \textbf{Question 13-30.} The accompanying data was extracted from the article ``Effects of Cold and Warm Temperatures on Springback of Aluminum-Magnesium Alloy 5083-H111'' (\textit{J. of Engr. Manuf.}, 2009: 427--431). The response variable is yield strength (MPa), and the predictor is temperature ($^\circ$ C).
 		\begin{center}
 		\begin{tabular}{c|c c c c c}
 			$x$ & -50 & 25 & 100 & 200 & 300 \\
 			$y$ & 91.0 & 120.5 & 136.0 & 133.1 & 120.8 \\
 		\end{tabular}
 		\end{center}
 		Minitab output is in the book from fitting the quadratic regression model (a graph in the cited paper suggests that the authors did this).
 		\begin{parts}
 			\part What proportion of observed variation in strength can be attributed to the model relationship?
 			\part Carry out a test of hypotheses at significance level .05 to decide if the quadratic predictor provides useful information over and above that provided by the linear predictor.
 			\part For a strength value of 100, $\hat{y} = 134.07, s_{\hat{Y}} = 2.38$. Estimate true average strength when temperature is 100, in a way that conveys information about precision and reliability.
 			\part Use the information in (c) to predict strength for a single observation to be made when temperature is 100, and do so in a way that conveys information about precision and reliability. Then compare this prediction to the estimate obtained in (c).
 		\end{parts}
 	\question \textbf{Question 13-36.} Cardiorespiratory fitness is widely recognized as a major component of overall physical well-being. Direct measurement of maximal oxygen uptake (VO$_2$max) is the single best measure of such fitness, but direct measurement is time-consuming and expensive. It is therefore desirable to have a prediction equation for VO$_2$max in terms of easily obtained quantities. Consider the variables
 		\begin{center}
 			$y$ = VO$_2$max (L/min) \\
 			$x)1$ = weight (kg) \\
 			$x_2$ = age(yr) \\
 			$x_3$ = time necessary to walk 1 mile (min) \\
 			$x_4$ = heart rate at the end of the walk (beats/min)
 		\end{center}
 		Here is one possible model, for male students, consistent with the information given in the article ``Validation of the Rockport Fitness Walking Test in College Males and Females'' (\textit{Research Quarterly for Exercise and Sport}, 1994: 152--158):
 		\begin{align*}
 			Y &= 5.0 + .01x_1 - .05x_2 - .13x_3 - .01x_4 + \epsilon \\
 			\sigma &= .4
 		\end{align*}
 		\begin{parts}
 			\part Interpret $\beta_1$ and $\beta_3$.
 			\part What is the expected value of VO$_2$max when weight is 76 kg, age is 20 yr, walk time is 12 min, and heart rate is 140 bpm?
 			\part What is the probability that VO$_2$max will be between 1.00 and 2.60 for a single observation made when the values of the predictors are as stated in part (b)?
 		\end{parts}
 	\question \textbf{Question 13-42.} An investigation of a die-casting process resulted in the accompanying data on $x_1$ = furnace temperature, $x_2$ = die close time, and $y$ = temperature difference on the die surface (``A Multiple-Objective Decision-Making Approach for Assessing Simultaneous Improvement in Die Life and Casting Quality in a Die Casting Process,'' \textit{Quality Engineering}, 1994: 371--383).
 		\begin{center}
 		\begin{tabular}{c|c c c c c c c c c}
 			\toprule
 			$x_1$ & 1250 & 1300 & 1350 & 1250 & 1300 & 1250 & 1300 & 1350 & 1350 \\
 			\midrule
 			$x_2$ & 6 & 7 & 6 & 7 & 6 & 8 & 8 & 7 & 8 \\
 			\midrule
 			$y$ & 80 & 95 & 101 & 85 & 92 & 87 & 96 & 106 & 108 \\
 			\bottomrule
 		\end{tabular}
 		\end{center}
 		Minitab output from fitting the multiple regression model with predictors $x_1$ and $x_2$ is given on page 569 of the textbook.
 		\begin{parts}
 			\part Carry out the model utility test.
 			\part Calculate and interpret a 95\% confidence interval for $\beta_2$, the population regression coefficient of $x_2$.
 			\part When $x_1 = 1300$ and $x_2 = 7$, the estimated standard deviation $\hat{Y}$ is $s_{\hat{Y}} = .353$. Calculate a 95\% confidence interval for true average temperature difference when furnace temperature is 1300 and die close time is 7.
 			\part Calculate a 95\% prediction interval for temperature difference resulting from a single experimental run with with a furnace temperature of 1300 and a die close time of 7.
 		\end{parts}
 	\end{questions}
 \end{document}

	\documentclass{exam}
	\usepackage[margin=1in]{geometry}
	\usepackage{amsmath}
	\usepackage{booktabs}
	\usepackage{listings}
	\usepackage{graphicx}
	\usepackage{paralist}
	\title{Statistics Homework 9}
	\author{Christian Mann}
	\date{\today}
	\begin{document}
	\maketitle
	\printanswers
	\begin{questions}
	\question \textbf{Question 12-36.} Mist (airborne droplets or aerosols) is generated when metal-removing fluids are used in machining operations to cool and lubricate the tool and workpiece. %heh, heh
	Mist generation is a concern to OSHA, which has recently lowered substantially the workplace standard. The article ``Variables Affecting Mist Generation from Metal Removal Fluids'' (\textit{Lubrication Engr.}, 2002: 10-17) gave the accompanying data on $x$ = fluid-flow velocity for a 5\% soluble oil (cm/sec) and $y$ = the extent of mist droplets having diameters smaller than 10$\mu$m (mg/m$^3$):
	\begin{center}
	\begin{tabular}{c \| c c c c c c c}
	\toprule
	$x$ & 89 & 177 & 189 & 354 & 362 & 442 & 965 \\
	\midrule
	$y$ & .40 & .60 & .48 & .66 & .61 & .69 & .99 \\
	\bottomrule
	\end{tabular}
	\end{center}
	\begin{parts}
	\part The investigators performed a simple linear regression analysis to relate the two variables. Does a scatter plot of the data support this strategy?
	\part What proportion of observed variation in mist can be attributed to the simple linear regression relationshipt between velocity and mist?
	\part The investigators were particularly interested in the impact on mist of increasing velocity from 100 to 1000 (a factor of 10 corresponding to difference between the smallest and largest $x$ values in the sample). When $x$ increases in this way, is there substantial evidence that the true average increase in $y$ is less than .6?
	\part Estimate the true average change in mist associated with a 1 cm/sec increase in velocity, and do so in a way that conveys information about precision and reliability.
	\end{parts}
	\question \textbf{Question 12-41.} \begin{parts}
	\part Verify that $E(\hat{\beta_1}) = \beta_1$ by using the rules of expected value from Chapter 5.
	\part Use the rules of variance from Chapter 5 to verify the expression for $V(\hat{\beta_1})$ given in this section.
	\end{parts}
	\question \textbf{Question 12-50.} Silicon-germanium alloys have been used in certain types of solar cells. The paper ``Silicon-Germanium Films Deposited by Low-Frequency Plasma-Enhanced Chemical Vapor Deposition'' (\textit{J. of Material Res.}, 2006: 88--104) reported on a study of various structural and electrical properties. Consider the accompanying data on $x$ = Ge concentration in solid phase (ranging from 0 to 1) and $y$ = Fermi level position (eV):
	\begin{center}
	\begin{tabular}{c\|c c c c c c c c c c c c c c}
	$x$ & 0 & .42 & .23 & .33 & .62 & .60 & .45 & .87 & .90 & .79 & 1 & 1 & 1 \\
	$y$ & .62 & .53 & .61 & .59 & .50 & .55 & .59 & .31 & .43 & .46 & .23 & .22 & .19 \\
	\end{tabular}
	\end{center}
	A scatter plot shows a substantial linear relationship. Here is Minitab output from a least squares fit. [\textit{Note:} There are several inconsistencies between the data given in the paper, the plot that appears there, and the summary information about a regression analysis.]
	\begin{verbatim}
	The regression equation is
	Fermi pos = 0.7217 - 0.4327 Ge conc

	S = 0.0737573 R-Sq = 80.2% R-Sq(adj) = 78.4%

	Analysis of Variance

	Source DF SS MS F P
	Regression 1 0.241728 0.241728 44.43 0.000
	Error 11 0.059842 0.005440
	Total 12 0.301569
	\end{verbatim}
	\begin{parts}
	\part Obtain an interval estimate of the expected change in Fermi-level position associated with an increase of .1 in Ge concentration, and interpret your result.
	\part Obtain an interval estimate for mean Fermi-level position when concentration is .50, and interpret your estimate.
	\part Obtain an interval of plausible values for position resulting from a single observation to be made when concentration is .50, interpret your interval, and compare to the interval of (b).
	\part Obtain simultaneous CIs for expected position when concentration is .3, .5, and .7; the joint confidence level should be at least 97\%.
	\end{parts}
	\question Plasma etching is essential to the fine-line pattern transfer in current semiconductor processes. The article ``Ion Beam-Assisted Etching of Aluminum with Chlorine'' (\textit{J. of the Electrochem. Soc.}, 1985: 2010--2012) gives the accompanying data (read from a graph) on chlorine flow ($x$, in SCCM) through a nozzle used in the eching mechanism and etch rate ($y$, in 100 A/min).
	\begin{center}
	\begin{tabular}{c\|c c c c c c c c c}
	\toprule
	$x$ & 1.5 & 1.5 & 2.0 & 2.5 & 2.5 & 3.0 & 3.5 & 3.5 & 4.0 \\
	\midrule
	$y$ & 23.0 & 24.5 & 25.0 & 30.0 & 33.5 & 40.0 & 40.5 & 47.0 & 49.0 \\
	\bottomrule
	\end{tabular}
	\end{center}
	The summary statistics are $\Sigma x_i = 24.0, \Sigma y_i = 312.5, \Sigma x_i^2 = 70.50, \Sigma x_iy_i = 902.5, \Sigma y_i^2 = 11,626.75, \hat{\beta_0} = 6.448718, \hat{\beta_1} = 10.602564$. \begin{parts}
	\part Does the simple linear regression model specify a useful relationshipt between chlorine flow and etch rate?
	\part Estimate the true average change in etch rate associated with a 1-SCCM increase in flow rate using a 95\% confidence interval, and interpret the interval.
	\part Calculate a 95\% CI for $\mu_{Y \cdot 3.0}$, the true average etch rate when flow = 3.0. Has this average been precisely estimated?
	\part Calculate a 95\% PI for a single future observation on etch rate to be made when flow = 3.0. Is the prediction likely to be accurate?
	\part Would the 95\% CI and PI when flow = 2.5 be wider or narrower than the corresponding intervals of parts (c) and (d)? Answer without actually computing the intervals.
	\part Would you recommend calculating a 95\% PI for a flow of 6.0? Explain.
	\end{parts}
	\question \textbf{Question 12-62.} Hydrogen content is conjectured to be an important factor in porosity of aluminum alloy castings. The article ``The Reduced Pressure Test as a Measuring Tool in the Evaluation of Porosity/Hydrogen Content in A1-7 Wt Pct Si-10 Vol Pct SiC(p) Metal Matrix Composite'' (\textit{Metallurgical Trans.}, 1993: 1857-1868) gives the accompanying data on $x$ = content and $y$ = gas porosity for one particular measurement technique.
	\begin{center}
	\begin{tabular}{c\|c c c c c c c c c c c c c c}
	\toprule
	$x$ & .18 & .20 & .21 & .21 & .21 & .22 & .23 & .23 & .24 & .24 & .25 & .28 & .30 & .37 \\
	\midrule
	$y$ & .46 & .70 & .41 & .45 & .55 & .44 & .24 & .47 & .22 & .80 & .88 & .70 & .72 & .75 \\
	\bottomrule
	\end{tabular}
	\end{center}
	Minitab gives the following output in response to a Correlation command:
	\begin{verbatim}
	Correlation of Hydrocon and Porosity = 0.449
	\end{verbatim}
	\begin{parts}
	\part Test at level .05 to see whether the population correlation coefficient differs from 0.
	\part If a simple linear regression analysis had been carried out, whate percentage of observed variation in porosity could be attributed to the model relationship?
	\end{parts}
	\question \textbf{Question 12-72.} The SAS output at the bottom of this page is based on data from the article ``Evidence for and the Rate of Denitrification in the Arabian Sea'' (\textit{Deep Sea Research}, 1978: 431-435). The variables under study are $x$ = salinity level (\%) and $y$ = nitrate level ($\mu$M/L).
	\begin{parts}
	\part What is the sample size $n$? [\textit{Hint}: Look for degrees of freedom for SSE.]
	\part Calculate a point estimate of expected nitrate level when salinity level is 35.5.
	\part Does there appear to be a useful linear relationship between the two variables?
	\part What is the value of the sample correlation coefficient?
	\part Would you use the simple linear regression model to draw conclusions when the salinity level is 40?
	\end{parts}
	\question \textbf{Question 13-20.} Exercise 14 presented data on body weight $x$ and metabolic clearance rate/body weight $y$. Consider the following intrinsically linear functions for specifying the relationship between the two variables: \begin{inparaenum}[(a)]
	\item ln($y$) versus $x$,
	\item ln($y$) versus ln($x$),
	\item $y$ versus ln($x$),
	\item $y$ versus 1/$x$, and
	\item ln($y$) versus 1/$x$
	\end{inparaenum}. Use any appropriate diagnostic plots and analyses to decide which of these functions you would select to specify a probabilistic model. Explain your reasoning.
	\question \textbf{Question 13-30.} The accompanying data was extracted from the article ``Effects of Cold and Warm Temperatures on Springback of Aluminum-Magnesium Alloy 5083-H111'' (\textit{J. of Engr. Manuf.}, 2009: 427--431). The response variable is yield strength (MPa), and the predictor is temperature ($^\circ$ C).
	\begin{center}
	\begin{tabular}{c\|c c c c c}
	$x$ & -50 & 25 & 100 & 200 & 300 \\
	$y$ & 91.0 & 120.5 & 136.0 & 133.1 & 120.8 \\
	\end{tabular}
	\end{center}
	Minitab output is in the book from fitting the quadratic regression model (a graph in the cited paper suggests that the authors did this).
	\begin{parts}
	\part What proportion of observed variation in strength can be attributed to the model relationship?
	\part Carry out a test of hypotheses at significance level .05 to decide if the quadratic predictor provides useful information over and above that provided by the linear predictor.
	\part For a strength value of 100, $\hat{y} = 134.07, s_{\hat{Y}} = 2.38$. Estimate true average strength when temperature is 100, in a way that conveys information about precision and reliability.
	\part Use the information in (c) to predict strength for a single observation to be made when temperature is 100, and do so in a way that conveys information about precision and reliability. Then compare this prediction to the estimate obtained in (c).
	\end{parts}
	\question \textbf{Question 13-36.} Cardiorespiratory fitness is widely recognized as a major component of overall physical well-being. Direct measurement of maximal oxygen uptake (VO$_2$max) is the single best measure of such fitness, but direct measurement is time-consuming and expensive. It is therefore desirable to have a prediction equation for VO$_2$max in terms of easily obtained quantities. Consider the variables
	\begin{center}
	$y$ = VO$_2$max (L/min) \\
	$x)1$ = weight (kg) \\
	$x_2$ = age(yr) \\
	$x_3$ = time necessary to walk 1 mile (min) \\
	$x_4$ = heart rate at the end of the walk (beats/min)
	\end{center}
	Here is one possible model, for male students, consistent with the information given in the article ``Validation of the Rockport Fitness Walking Test in College Males and Females'' (\textit{Research Quarterly for Exercise and Sport}, 1994: 152--158):
	\begin{align*}
	Y &= 5.0 + .01x_1 - .05x_2 - .13x_3 - .01x_4 + \epsilon \\
	\sigma &= .4
	\end{align*}
	\begin{parts}
	\part Interpret $\beta_1$ and $\beta_3$.
	\part What is the expected value of VO$_2$max when weight is 76 kg, age is 20 yr, walk time is 12 min, and heart rate is 140 bpm?
	\part What is the probability that VO$_2$max will be between 1.00 and 2.60 for a single observation made when the values of the predictors are as stated in part (b)?
	\end{parts}
	\question \textbf{Question 13-42.} An investigation of a die-casting process resulted in the accompanying data on $x_1$ = furnace temperature, $x_2$ = die close time, and $y$ = temperature difference on the die surface (``A Multiple-Objective Decision-Making Approach for Assessing Simultaneous Improvement in Die Life and Casting Quality in a Die Casting Process,'' \textit{Quality Engineering}, 1994: 371--383).
	\begin{center}
	\begin{tabular}{c\|c c c c c c c c c}
	\toprule
	$x_1$ & 1250 & 1300 & 1350 & 1250 & 1300 & 1250 & 1300 & 1350 & 1350 \\
	\midrule
	$x_2$ & 6 & 7 & 6 & 7 & 6 & 8 & 8 & 7 & 8 \\
	\midrule
	$y$ & 80 & 95 & 101 & 85 & 92 & 87 & 96 & 106 & 108 \\
	\bottomrule
	\end{tabular}
	\end{center}
	Minitab output from fitting the multiple regression model with predictors $x_1$ and $x_2$ is given on page 569 of the textbook.
	\begin{parts}
	\part Carry out the model utility test.
	\part Calculate and interpret a 95\% confidence interval for $\beta_2$, the population regression coefficient of $x_2$.
	\part When $x_1 = 1300$ and $x_2 = 7$, the estimated standard deviation $\hat{Y}$ is $s_{\hat{Y}} = .353$. Calculate a 95\% confidence interval for true average temperature difference when furnace temperature is 1300 and die close time is 7.
	\part Calculate a 95\% prediction interval for temperature difference resulting from a single experimental run with with a furnace temperature of 1300 and a die close time of 7.
	\end{parts}
	\end{questions}
	\end{document}