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Applied Statistics and Probability for Engineers

Douglas C. Montgomery, George C. Runger

Chapter 12

Multiple Linear Regression - all with Video Answers

Educators

Section 1

Multiple Linear Regression Model

05:27

Problem 1

A study was performed to investigate the shear strength of soil $(y)$ as it related to depth in feet $\left(x_{1}\right)$ and $\%$ moisture content $\left(x_{2}\right)$. Ten observations were collected, and the following summary quantities obtained: $n=10, \sum x_{i 1}=223,$ $\sum x_{2}=553, \Sigma y_{i}=1,916, \sum x_{i 1}^{2}=5,200.9, \sum x_{R}^{2}=31,729$ $\sum x_{i 1} x_{i 2}=12,352, \sum x_{i 1} y_{i}=43,550.8, \sum x_{i 2} y_{i}=104,736.8$ and $\sum y_{i}^{2}=371,595.6$
(a) Set up the least squares normal equations for the model $Y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\epsilon$
(b) Estimate the parameters in the model in part (a).
(c) What is the predicted strength when $x_{1}=18$ feet and $x_{2}=43 \% ?$

Jameson Kuper
Jameson Kuper
Numerade Educator
07:01

Problem 2

A regression model is to be developed for predicting the ability of soil to absorb chemical contaminants. Ten observations have been taken on a soil absorption index $(y)$ and two regressors: $x_{1}=$ amount of extractable iron ore and $x_{2}=$ amount of bauxite. We wish to fit the model $Y=\beta_{0}+\beta_{1} x_{1}+$ $\beta_{2} x_{2}+\epsilon$. Some necessary quantities are.
$$ \begin{array}{c} \left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1}=\left[\begin{array}{ccc} 1.17991 & -7.30982 \mathrm{E}-3 & 7.3006 \mathrm{E}-4 \\ -7.30982 \mathrm{E}-3 & 7.9799 \mathrm{E}-5 & -1.23713 \mathrm{E}-4 \\ 7.3006 \mathrm{E}-4 & -1.23713 \mathrm{E}-4 & 4.6576 \mathrm{E}-4 \end{array}\right] \\ \mathbf{X}^{\prime} \mathbf{y}=\left[\begin{array}{r} 220 \\ 36,768 \\ 9,965 \end{array}\right] \end{array} $$
(a) Estimate the regression coefficients in the model specified above.
(b) What is the predicted value of the absorption index $y$ when $x_{1}=200$ and $x_{2}=50 ?$

Heather Zimmers
Heather Zimmers
Numerade Educator
01:24

Problem 3

A chemical engineer is investigating how the amount of conversion of a product from a raw material $(y)$ depends on reaction temperature $\left(x_{1}\right)$ and the reaction time $\left(x_{2}\right) .$ He has developed the following regression models:
$$ \begin{array}{l} \text { 1. } \hat{y}=100+2 x_{1}+4 x_{2} \\ \text { 2. } \hat{y}=95+1.5 x_{1}+3 x_{2}+2 x_{1} x_{2} \end{array} $$
Both models have been built over the range $0.5 \leq x_{2} \leq 10$.
(a) What is the predicted value of conversion when $x_{2}=2 ?$ Repeat this calculation for $x_{2}=8 .$ Draw a graph of the predicted values for both conversion models. Comment on the effect of the interaction term in model 2 .
(b) Find the expected change in the mean conversion for a unit change in temperature $x_{1}$ for model 1 when $x_{2}=5 .$ Does this quantity depend on the specific value of reaction time selected? Why?
(c) Find the expected change in the mean conversion for a unit change in temperature $x_{1}$ for model 2 when $x_{2}=5 .$ Repeat this calculation for $x_{2}=2$ and $x_{2}=8$. Does the result depend on the value selected for $x_{2}$ ? Why?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
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Problem 4

$12.4 .$ You have fit a multiple linear regression model and the $\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1}$ matrix is:
$$ \left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1}=\left[\begin{array}{rrr} 0.893758 & -0.0282448 & -0.0175641 \\ -0.028245 & 0.0013329 & 0.0001547 \\ -0.017564 & 0.0001547 & 0.0009108 \end{array}\right] $$
(a) How many regressor variables are in this model?
(b) If the error sum of squares is 307 and there are 15 observations, what is the estimate of $\sigma^{2} ?$
(c) What is the standard error of the regression coefficient $\hat{\beta}_{1} ?$

Rashmi Sinha
Rashmi Sinha
Numerade Educator
07:16

Problem 5

12.5. Data from a patient satisfaction survey in a hospital are shown in the following table:
The regressor variables are the patient's age, an illness severity index (larger values indicate greater severity), an indicator variable denoting whether the patient is a medical patient (0) or a surgical patient (1), and an anxiety index (larger values indicate greater anxiety).
(a) Fit a multiple linear regression model to the satisfaction response using age, illness severity, and the anxiety index as the regressors.
(b) Estimate $\sigma^{2}$
(c) Find the standard errors of the regression coefficients.
(d) Are all of the model parameters estimated with nearly the same precision? Why or why not?

Jerelyn Nevil
Jerelyn Nevil
Numerade Educator
03:35

Problem 6

The electric power consumed each month by a chemical plant is thought to be related to the average ambient temperature $\left(x_{1}\right),$ the number of days in the month $\left(x_{2}\right),$ the average product purity $\left(x_{3}\right),$ and the tons of product produced $\left(x_{4}\right) .$ The past year's historical data are available and are presented in the following table:
$$ \begin{array}{ccccc} \hline y & x_{1} & x_{2} & x_{3} & x_{4} \\ \hline 240 & 25 & 24 & 91 & 100 \\ 236 & 31 & 21 & 90 & 95 \\ 270 & 45 & 24 & 88 & 110 \\ 274 & 60 & 25 & 87 & 88 \\ 301 & 65 & 25 & 91 & 94 \\ 316 & 72 & 26 & 94 & 99 \\ 300 & 80 & 25 & 87 & 97 \\ 296 & 84 & 25 & 86 & 96 \\ 267 & 75 & 24 & 88 & 110 \\ 276 & 60 & 25 & 91 & 105 \\ 288 & 50 & 25 & 90 & 100 \\ 261 & 38 & 23 & 89 & 98 \\ \hline \end{array} $$(a) Fit a multiple linear regression model to these data.
(b) Estimate $\sigma^{2}$
(c) Compute the standard errors of the regression coefficients. Are all of the model parameters estimated with the same precision? Why or why not?
(d) Predict power consumption for a month in which $x_{1}=75^{\circ} \mathrm{F}, x_{2}=24$ days, $x_{3}=90 \%,$ and $x_{4}=98$ tons.

Hunza Gilgit
Hunza Gilgit
Numerade Educator
03:39

Problem 7

Table $12-5$ provides the highway gasoline mileage test results for 2005 model year vehicles from DaimlerChrysler. The full table of data (available on the book's Web site) contains the same data for 2005 models from over 250 vehicles from many manufacturers (source: Environmental Protection Agency Web site www.epa.gov/ otag/cert/mpg/testcars/database).
(a) Fit a multiple linear regression model to these data to estimate gasoline mileage that uses the following regressors:
cid, $r h p,$ etw, $c m p,$ axle, $n / v$ (b) Estimate $\sigma^{2}$ and the standard errors of the regression coefficients.
(c) Predict the gasoline mileage for the first vehicle in the table.

Bon Zapata
Bon Zapata
Numerade Educator
07:28

Problem 8

The pull strength of a wire bond is an important characteristic. The following table gives information on pull strength $(y),$ die height $\left(x_{1}\right),$ post height $\left(x_{2}\right),$ loop height $\left(x_{3}\right)$ wire length $\left(x_{4}\right)$, bond width on the die $\left(x_{5}\right)$, and bond width on the post $\left(x_{6}\right)$.
(a) Fit a multiple linear regression model using $x_{2}, x_{3}, x_{4},$ and $x_{5}$ as the regressors.
(b) Estimate $\sigma^{2}$(d) Use the model from part (a) to predict pull strength when $x_{2}=20, x_{3}=30, x_{4}=90,$ and $x_{5}=2.0$
(c) Find the $\operatorname{se}\left(\hat{\beta}_{j}\right)$. How precisely are the regression coefficients estimated, in your opinion?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
11:56

Problem 9

An engineer at a semiconductor company wants to model the relationship between the device HFE $(y)$ and three parameters: Emitter-RS $\left(x_{1}\right),$ Base-RS $\left(x_{2}\right),$ and Emitter-to-Base $\operatorname{RS}\left(x_{3}\right) .$ The data are shown in the following table. (a) Fit a multiple linear regression model to the data.
(b) Estimate $\sigma^{2}$
(c) Find the standard errors $\operatorname{se}\left(\hat{\beta}_{j}\right)$. Are all of the model parameters estimated with the same precision? Justify your answer.
(d) Predict HFE when $x_{1}=14.5, x_{2}=220,$ and $x_{3}=5.0$.

Srikar Katta
Srikar Katta
Numerade Educator
03:01

Problem 10

Heat treating is often used to carburize metal parts, such as gears. The thickness of the carburized layer is considered a crucial feature of the gear and contributes to the overall reliability of the part. Because of the critical nature of this feature, two different lab tests are performed on each furnace load. One test is run on a sample pin that accompanies each load. The other test is a destructive test, where an actual part is cross-sectioned. This test involves running a carbon analysis on the surface of both the gear pitch (top of the gear tooth) and the gear root (between the gear teeth). Table $12-6$ shows the results of the pitch carbon analysis test for 32 parts.
The regressors are furnace temperature (TEMP), carbon concentration and duration of the carburizing cycle (SOAKPCT, SOAKTIME), and carbon concentration and duration of the diffuse cycle (DIFFPCT, DIFFTIME).
(a) Fit a linear regression model relating the results of the pitch carbon analysis test (PITCH) to the five regressor variables.
(b) Estimate $\sigma^{2}$
(c) Find the standard errors $\operatorname{se}\left(\dot{\beta}_{j}\right)$.
(d) Use the model in part (a) to predict PITCH when $\mathrm{TEMP}=1650,$ SOAKTIME $=1.00, \mathrm{SOAKPCT}=1.10$
DIFFTIME $=1.00$, and DIFFPCT $=0.80$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:20

Problem 11

An article in Electronic Packaging and Production (2002, Vol. 42) considered the effect of X-ray inspection of integrated circuits. The rads (radiation dose) were studied as a function of current (in milliamps) and exposure time (in minutes).
(a) Iit a multiple linear regression model to these data with rads as the response
(b) Estimate $\sigma^{2}$ and the standard errors of the regression coefficients.
(c) Use the model to predict rads when the current is 15 milliamps and the exposure time is 5 seconds.

JD
Jhamku Devi
Numerade Educator
02:31

Problem 12

An article in Cancer Epidemiology, Biomarkers and Prevention $(1996,$ Vol. $5,$ pp. $849-852)$ conducted a pilot study to assess the use of toenail arsenic concentrations as an indicator of ingestion of arsenic-containing water. Twenty-one participants were interviewed regarding use of their private (unregulated) wells for drinking and cooking, and each provided a sample of water and toenail clippings. The table below showed the data of age (years), sex of person $(1=$ male, $2=$ female), proportion of times household well used for drinking $(1 \leq 1 / 4,2=1 / 4,3=1 / 2,4=3 / 4,5 \geq 3 / 4),$ proportion of times household well used for cooking $(1 \leq 1 / 4,2=1 / 4,3=$ $1 / 2,4=3 / 4,5 \geq 3 / 4),$ arsenic in water $(\mathrm{ppm}),$ and arsenic in toenails (ppm) respectively. (a) lit a multiple linear regression model using arsenic concentration in nails as the response and age, drink use, cook usc, and arsenic in the water as the regressors.
(b) Estimate $\sigma^{2}$ and the standard errors of the regression coefficients.
(c) Use the model to predict the arsenic in nails when the age is $30,$ the drink use is catcgory $5,$ the cook use is catcgory $5,$ and arsenic in the water is $0,135 \mathrm{ppm}$.

Lucas Finney
Lucas Finney
Numerade Educator
02:37

Problem 13

In an article in /EFE Transactions on Instrumentation and Measurement $(2001,$ Vol $.50,$ pp. $2033-2040)$ powdered mixtures of coal and limestone were analyzed for permittivity. The crrors in the density measurement was the response.(a) Fit a multiple linear regression model to these data with the density as the response.
(b) Estimate $\sigma^{2}$ and the standard errors of the regression coefficients.
(c) Use the model to predict the density when the dielectric constant is 2.5 and the loss factor is $0.03 .$

Raymond Matshanda
Raymond Matshanda
Numerade Educator
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Problem 14

An article in Biotechnology Progress (2001, Vol. $17,$ pp. $366-368$ ) reported on an experiment to investigate and optimize nisin extraction in aqueous two-phase systems (ATPS). The nisin recovery was the dependent variable ( $y$ ). The two regressor variables were concentration (\%) of PEG $4000\left(\right.$ denoted as $\left.x_{1}\right)$ and concentration $(\%)$ of $\mathrm{Na}_{2} \mathrm{SO}_{4}($ denoted as $x_{2}$ ).
(a) Fit a multiple linear regression model to these data.
(b) Estimate $\sigma^{2}$ and the standard errors of the regression coefficients.
(c) Use the model to predict the nisin recovery when $x_{1}=14.5$ and $x_{2}=12.5$

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:19

Problem 15

An article in Optical Engineering ["Operating Curve Extraction of a Corrclator's Filter" (2004, Vol. $43,$ pp. $2775-2779)]$ reported on use of an optical correlator to perform an experiment by varying brightness and contrast. The resulting modulation is characterized by the useful range of gray levels. The data are shown below:
$\begin{array}{llllrrrrr}\text { Brightncss (\%): } & 54 & 61 & 65 & 100 & 100 & 100 & 50 & 57 & 54 \\ \text { Contrast (\%): } & 56 & 80 & 70 & 50 & 65 & 80 & 25 & 35 & 26 \\ \text { Useful range (ng): } & 96 & 50 & 50 & 112 & 96 & 80 & 155 & 144 & 255\end{array}$
(a) lit a multiple linear regression model to these data.
(b) Iistimate $\sigma^{2}$
(c) Compute the standard errors of the regression coefficicnts.
(d) Prediet the uscful range when brightness $=80$ and contrast $=75$

Sana Riaz
Sana Riaz
Numerade Educator
08:01

Problem 16

An article in Technometrics $(1974,$ Vol. $16,$ pp. $523-531$ ) considered the following stack-loss data from a plant oxidizing ammonia to nitric acid. Twenty-one daily responses of stack loss $y$ (the amount of ammonia escaping) were measured with air flow $x_{1},$ temperature $x_{2},$ and acid conccntration $x_{3}$ $ \begin{aligned} \text { Stack } \operatorname{loss} y=42,37,37,28,18,18,19,20,15,14,14,13, \\ & 11,12,8,7,8,8,9,15,15 \\ x_{1}=80,80,75,62,62,62,62,62,58,58,58,58,58,58,50, \\
50,50,50,50,56,70 \end{aligned} $
(a) Fit a linear regression model relating the results of the stack loss to the three regressor varilables.
(b) Estimate $\sigma^{2}$.
(c) Find the standard error $\operatorname{se}\left(\hat{\beta}_{j}\right)$.
(d) Use the model in part (a) to predict stack loss when $x_{1}=60$, $x_{2}=26,$ and $x_{3}=85$

Heather Duong
Heather Duong
Numerade Educator
04:44

Problem 17

Table $12-7$ presents quarterback ratings for the 2008 National Football League season (source: The Sports Network).
(a) Fit a multiple regression model to relate the quarterback rating to the percentage of completions, the percentage of TDs, and the percentage of interceptions.
(b) Fstimate $\boldsymbol{\sigma}^{2}$.
(c) What are the standard errors of the regression coefficients?
(d) Use the model to predict the rating when the percentage of completions is $60 \%,$ the percentage of TI $) \mathrm{s}$ is $4 \%,$ and the percentage of interceptions is $3 \%$.

Jameson Kuper
Jameson Kuper
Numerade Educator
03:35

Problem 18

Table $12-8$ presents statistics for the National Hockey League teams from the $2008-2009$ season (source: The Sports Network). Fit a multiple linear regression model that relates Wins to the variables GF through $F G$. Because teams play 82 games $W=82-L-T-O T L,$ but such a model docs not help build a better team. Estimate $\sigma^{2}$ and find the standard errors of the regression coefficients for your model.

Heena Haldankar
Heena Haldankar
Numerade Educator
04:08

Problem 19

A study was performed on wear of a bearing $y$ and its relationship to $x_{1}=$ oil viscosity and $x_{2}=$ loud. The following data were obtained.$\begin{array}{rrr} \hline y & x_{1} & x_{2} \\ \hline 293 & 1.6 & 851 \\ 230 & 15.5 & 816 \\ 172 & 22.0 & 1058 \\
91 & 43.0 & 1201 \\ 113 & 33.0 & 1357 \\ 125 & 40.0 & 1115 \\ \hline \end{array} $
(a) Iit a multiple linear regression model to these data
(b) Iistimate $\sigma^{2}$ and the standard errors of the regression coefficicnts.
(c) Use the model to prodict wear when $x_{1}=25$ and $x_{2}=1000$.
(d) Fit a multiple linear regression model with an interaction term to these data
(c) Estimate $\sigma^{2}$ and $\operatorname{se}\left(\hat{\beta}_{j}\right)$ for this ncw model. How did these quantitics change? Does this tell you anything about the value of adding the interaction term to the model?
(f) Use the model in (d) to predict when $x_{1}=25$ and $x_{2}=$
1000. Compare this prediction with the predicted value from part (c) above.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:33

Problem 20

Consider the linear regression model
$$ Y_{j}=\beta_{0}^{\prime}+\beta_{1}\left(x_{i 1}-\bar{x}_{1}\right)+\beta_{2}\left(x_{i 2}-\bar{x}_{2}\right)+\epsilon_{i} $$
where $\bar{x}_{1}=\sum x_{i 1} / n$ and $\bar{x}_{2}=\sum x_{i} / n .$
(a) Write out the least squares normal equations for this model.
(b) Verify that the least squares estimate of the intercept in this model is $\hat{\beta}_{0}^{\prime}=\sum y_{d} / n=\bar{y}$
(c) Suppose that we use $y_{i}-\bar{y}$ as the response variable in the model above What effect will this have on the least squares estimate of the intercept?

James Kiss
James Kiss
Numerade Educator