• Home
  • Textbooks
  • Mathematical Statistics and Data Analysis
  • Linear Least Squares

Mathematical Statistics and Data Analysis

John A. Rice

Chapter 14

Linear Least Squares - all with Video Answers

Educators

+ 2 more educators

Chapter Questions

View

Problem 1

Convert the following relationships into linear relationships by making transformations and defining new variables.
a. $y=a /(b+c x)$
b. $y=a e^{-b x}$
c. $y=a b^{x}$
d. $y=x /(a+b x)$
e. $y=1 /\left(1+e^{b x}\right)$

Rashmi Sinha
Rashmi Sinha
Numerade Educator
03:53

Problem 2

Plot $y$ versus $x$ for the following pairs:
$$\begin{array}{c|cccccccccc}
x & 34 & 1.38 & -.65 & .68 & 1.40 & -.88 & -.30 & -1.18 & 50 & -1.75 \\
\hline y & .27 & 1.34 & -.53 & .35 & 1.28 & -.98 & -.72 & -.81 & .64 & -1.59
\end{array}$$
a. Fit a line $y=a+b x$ by the method of least squares, and sketch it on the plot.
b. Fit a line $x=c+d y$ by the method of least squares, and sketch it on the plot.
c. Are the lines in parts (a) and (b) the same? If not, why not?

Jameson Kuper
Jameson Kuper
Numerade Educator
04:10

Problem 3

Suppose that $y_{i}=\mu+e_{i},$ where $i=1, \ldots ., n$ and the $e_{i}$ are independent errors with mean zero and variance $\sigma^{2}$. Show that $\bar{y}$ is the least squares estimate of $\mu$.

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
View

Problem 4

Consider a standard linear regression model in which the freshman GPA is modeled to depend linearly on high school GPA: $Y_{i}=\beta_{0}+\beta_{1} x_{i}+e_{i}, i=1,2 \ldots, n$ Suppose that different intercepts were to be allowed for females and males, and write the model as
$$
Y_{i}=I_{F}(i) \beta_{F}+I_{M}(i) \beta_{M}+\beta_{1} x_{i}+e_{i}
$$
where $I_{F}(i)$ and $I_{M}(i)$ are indicator variables taking on values 0 and 1 according to whether the gender of the $i$ th person is female or male. Give the form of the design matrix for such a model.

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 5

Three objects are located on a line at points $p_{1}<p_{2}<p_{3} .$ These locations are not precisely known. A surveyor makes the following measurements:
a. He stands at the origin and measures the three distances from there to $p_{1}, p_{2}$ $p_{3} .$ Let these measurements be denoted by $Y_{1}, Y_{2}, Y_{3}$
b. He goes to $p_{1}$ and measures the distances from there to $p_{2}$ and $p_{3}$. Let these measurements be denoted by $Y_{4}, Y_{5}$
c. He goes to $p_{2}$ and measures the distance from there to $p_{3}$. Denote this measurement by $Y_{6}$ He thus makes six measurements in all, and they are all subject to error. In order to estimate the values $p_{1}, p_{2}, p_{3},$ he decides to combine all the measurements
by the method of least squares. Using matrix notation, explain clearly how the least squares estimates would be calculated (you don't have to do the actual calculations).

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 6

Three objects are located on a line at points $p_{1}<p_{2}<p_{3} .$ These locations are not precisely known. A surveyor makes the following measurements:
a. He stands at the origin and measures the three distances from there to $p_{1}, p_{2}$ $p_{3} .$ Let these measurements be denoted by $Y_{1}, Y_{2}, Y_{3}$
b. He goes to $p_{1}$ and measures the distances from there to $p_{2}$ and $p_{3}$. Let these measurements be denoted by $Y_{4}, Y_{5}$
c. He goes to $p_{2}$ and measures the distance from there to $p_{3}$. Denote this measurement by $Y_{6}$ He thus makes six measurements in all, and they are all subject to error. In order to estimate the values $p_{1}, p_{2}, p_{3},$ he decides to combine all the measurements
by the method of least squares. Using matrix notation, explain clearly how the least squares estimates would be calculated (you don't have to do the actual calculations).
c. Find the estimate of $\sigma^{2}$
d. Find the estimated standard errors of the least squares estimates of part (b).
e. Estimate $w_{1}-w_{2}$ and its standard error.
f. Test the null hypothesis $H_{0}: w_{1}=w_{2}$

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 7

(Weighted Least Squares) Suppose that in the model $y_{i}=\beta_{0}+\beta_{1} x_{i}+e_{i},$ the errors have mean zero and are independent, but $\operatorname{Var}\left(e_{i}\right)=\rho_{i}^{2} \sigma^{2},$ where the $\rho_{i}$ are known constants, so the errors do not have equal variance. This situation arises when the $y_{i}$ are averages of several observations at $x_{i}$; in this case, if $y_{i}$
is an average of $n_{i}$ independent observations, $\rho_{i}^{2}=1 / n_{i}$ (why?). Because the variances are not equal, the theory developed in this chapter does not apply; intuitively, it seems that the observations with large variability should influence the estimates of $\beta_{0}$ and $\beta_{1}$ less than the observations with small variability. The problem may be transformed as follows:
$$
\rho_{i}^{-1} y_{i}=\rho_{i}^{-1} \beta_{0}+\rho_{i}^{-1} \beta_{1} x_{i}+\rho_{i}^{-1} e_{i}
$$
or
$$
z_{i}=u_{i} \beta_{0}+v_{i} \beta_{1}+\delta_{i}
$$
where
$$
u_{i}=\rho_{i}^{-1} \quad v_{i}=\rho_{i}^{-1} x_{i} \quad \delta_{i}=\rho_{i}^{-1} e_{i}
$$
a. Show that the new model satisfies the assumptions of the standard statistical model.
b. Find the least squares estimates of $\beta_{0}$ and $\beta_{1}$
c. Show that performing a least squares analysis on the new model, as was done
in part (b), is equivalent to minimizing
$$
\sum_{i=1}^{n}\left(y_{i}-\beta_{0}-\beta_{1} x_{i}\right)^{2} \rho_{i}^{-2}
$$
This is a weighted least squares criterion; the observations with large variances are weighted less.
d. Find the variances of the estimates of part (b).

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 8

(The QR Method) This problem outlines the basic ideas of an alternative method, the QR method, of finding the least squares estimate $\hat{\boldsymbol{\beta}}$. An advantage of the method is that it does not include forming the matrix $\mathbf{X}^{T} \mathbf{X},$ a process that tends to increase rounding error. The essential ingredicnt of the method is that if $\mathbf{X}_{n} \times p$ has $p$ linearly independent columns, it may be factored in the form
$$
\mathbf{x}=\mathbf{Q} \quad \mathbf{R}
$$
$m \times p \quad m \times p p \times p$
where the columns of $\mathbf{Q}$ are orthogonal $\left(\mathbf{Q}^{T} \mathbf{Q}=\mathbf{I}\right)$ and $\mathbf{R}$ is upper-triangular $\left(r_{i j}=0, \text { for } i>j\right)$ and nonsingular. [For a discussion of this decomposition and its relationship to the Gram-Schmidt process, see Strang (1980).] Show that $\hat{\boldsymbol{\beta}}=\left(\mathbf{X}^{T} \mathbf{X}\right)^{-1} \mathbf{X}^{T} \mathbf{Y}$ may also be expressed as $\hat{\boldsymbol{\beta}}=\mathbf{R}^{-1} \mathbf{Q}^{T} \mathbf{Y}$
or $\mathbf{R} \hat{\boldsymbol{\beta}}=\mathbf{Q}^{T} \mathbf{Y} .$ Indicate how this last equation may be solved for $\hat{\boldsymbol{\beta}}$ by backsubstitution, using that $\mathbf{R}$ is upper-triangular, and show that it is thus unnecessary
to invert $\mathbf{R}$

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 9

(Cholesky Decomposition) This problem outlines the basic ideas of a popular and effective method of computing least squares estimates. Assuming that its inverse exists, $\mathbf{X}^{T} \mathbf{X}$ is a positive, definite matrix and may be factored as $\mathbf{X}^{T} \mathbf{X}=\mathbf{R}^{T} \mathbf{R}$ where $\mathbf{R}$ is an upper-triangular matrix. This factorization is called the Cholesky decomposition. Show that the least squares estimates can be found by solving the equations
$$
\begin{aligned}
\mathbf{R}^{T} \mathbf{v} &=\mathbf{x}^{T} \mathbf{Y} \\
\mathbf{R} \hat{\boldsymbol{\beta}} &=\mathbf{v}
\end{aligned}
$$
where $v$ is appropriately defined. Show that these equations can be solved by back-substitution because $\mathbf{R}$ is upper-triangular, and that therefore it is not necessary to carry out any matrix inversions explicitly to find the least squares estimates.

Victor Salazar
Victor Salazar
Numerade Educator
02:16

Problem 10

Show that the least squares estimates of the slope and intercept of a line may be expressed as
$$
\beta_{0}=\bar{y}-\beta_{1} \bar{x}
$$
and
$$
\hat{\beta}_{1}=\frac{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)\left(y_{i}-\bar{y}\right)}{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}}
$$

Rashmi Sinha
Rashmi Sinha
Numerade Educator
View

Problem 11

Show that if $\bar{x}=0,$ the estimated slope and intercept are uncorrelated under the assumptions of the standard statistical model.

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 12

Use the result of Problem 10 to show that the line fit by the method of least squares passes through the point $(\bar{x}, \bar{y})$

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 13

Suppose that a line is fit by the method of least squares to $n$ points, that the standard statistical model holds, and that we want to estimate the line at a new point, $x_{0} .$ Denoting the value on the line by $\mu_{0},$ the estimate is
$$
\hat{\mu}_{0}=\beta_{0}+\beta_{1} x_{0}
$$
a. Derive an expression for the variance of $\mu_{0}$
b. Sketch the standard deviation of $\hat{\mu}_{0}$ as a function of $x_{0}-\bar{x}$. The shape of the curve should be intuitively plausible.
c. Derive a $95 \%$ confidence interval for $\mu_{0}=\beta_{0}+\beta_{1} x_{0}$ under an assumption of normality.

Shu Naito
Shu Naito
Numerade Educator
View

Problem 14

Problem 13 dealt with how to form a confidence interval for the value of a line at a point $x_{0} .$ Suppose that instead we want to predict the value of a new observation, $Y_{0},$ at $x_{0}$
$$
Y_{0}=\beta_{0}+\beta_{1} x_{0}+e_{0}
$$
by the estimate
$$
\hat{Y}_{0}=\hat{\beta}_{0}+\hat{\beta}_{1} x_{0}
$$
a. Find an expression for the variance of $\hat{Y}_{0}-Y_{0},$ and compare it to the expression for the variance of $\hat{\mu}_{0}$ obtained in part (a) of Problem $13 .$ Assume that $e_{0}$ is independent of the original observations and has the variance $\sigma^{2}$
b. Assuming that $e_{0}$ is normally distributed, find the distribution of $Y_{0}-Y_{0} .$ Use this result to find an interval $I$ such that $P\left(Y_{0} \in I\right)=1-\alpha$. This interval is called a $100(1-\alpha) \%$ prediction interval.

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 15

Find the least squares estimate of $\beta$ for fitting the line $y=\beta x$ to points $\left(x_{i}, y_{i}\right)$ where $i=1, \ldots, n$

Victor Salazar
Victor Salazar
Numerade Educator
View

Problem 16

Consider fitting the curve $y=\beta_{0} x+\beta_{1} x^{2}$ to points $\left(x_{i}, y_{i}\right),$ where $i=1, \ldots, n$
a. Use the matrix formalism to find expressions for the least squares estimates of $\beta_{0}$ and $\beta_{1}$
b. Find an expression for the covariance matrix of the estimates.

Victor Salazar
Victor Salazar
Numerade Educator
03:38

Problem 17

This problem extends some of the material in Section $14.2 .3 .$ Let $X$ and $Y$ be random variables with
$$
\begin{array}{c}
E(X)=\mu_{x} \quad E(Y)=\mu_{y} \\
\operatorname{Var}(X)=\sigma_{x}^{2} \quad \operatorname{Var}(Y)=\sigma_{y}^{2} \\
\operatorname{Cov}(X, Y)=\sigma_{x y}
\end{array}
$$
Consider predicting $Y$ from $X$ as $\hat{Y}=\alpha+\beta X,$ where $\alpha$ and $\beta$ are chosen to minimize $E(Y-\tilde{Y})^{2},$ the expected squared prediction error.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
View

Problem 18

Suppose that
$$
Y_{i}=\beta_{0}+\beta_{1} x_{i}+e_{i}, \quad i=1, \ldots, n
$$
where the $e_{i}$ are independent and normally distributed with mean zero and variance $\sigma^{2}$. Find the mle's of $\beta_{0}$ and $\beta_{1}$ and verify that they are the least squares estimates. (Hint: Under these assumptions, the $Y_{i}$ are independent and normally distributed with means $\beta_{0}+\beta_{1} x_{i}$ and variance $\sigma^{2}$. Write the joint density function of the $Y_{i}$ and thus the likelihood.)

Victor Salazar
Victor Salazar
Numerade Educator
01:00

Problem 19

a. Show that the vector of residuals is orthogonal to every column of $\mathbf{X}$.
b. Use this result to show that the residuals sum to zero and thus the sum has expectation zero if the model contains an intercept term.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:00

Problem 20

a. Show that the vector of residuals is orthogonal to every column of $\mathbf{X}$.
b. Use this result to show that the residuals sum to zero and thus the sum has expectation zero if the model contains an intercept term.

Manik Pulyani
Manik Pulyani
Numerade Educator
02:53

Problem 21

Assume that the columns of $\mathbf{X}, \mathbf{X}_{1}, \ldots, \mathbf{X}_{p},$ are orthogonal; that is, $\mathbf{X}_{i}^{T} \mathbf{X}_{j}=0$ for $i \neq j .$ Show that the covariance matrix of the least squares estimates is diagonal.

Jack Chen
Jack Chen
Numerade Educator
View

Problem 22

Suppose that the relation of family income to consumption is linear. Of those families in the 90 th percentile of income, what proportion would you expect to be at or above the 90 th percentile of consumption: (a) exactly $50 \%,$ (b) less than
$50 \%,$ (c) more than $50 \% ?$ Justify your answers.

Shu Naito
Shu Naito
Numerade Educator
03:14

Problem 23

Suppose that grades on a midterm and a final have a correlation coefficient of . 5 and both exams have an average score of 75 and a standard deviation of $10 .$
a. If a student's score on the midterm is $95,$ what would you predict his score on the final to be?
b. If a student scored 85 on the final, what would you guess that her score on the midterm was?

Jameson Kuper
Jameson Kuper
Numerade Educator
05:55

Problem 24

Suppose that the independent variables in a least squares problem are replaced by rescaled variables $u_{i j}=k_{j} x_{i j}$ (for example, centimeters are converted to meters.) Show that $Y$ does not change. Does $\hat{\beta}$ change? (Hint: Express the new design matrix in terms of the old one.)

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
View

Problem 25

Suppose that each setting $x_{i}$ of the independent variables in a simple least squares problem is duplicated, yielding two independent observations $Y_{i_{1}}, Y_{i_{i}},$ Is it true that the least squares estimates of the intercept and slope can be found by doing a regression of the mean responses of each pair of duplicates, $\bar{Y}_{i}=\left(Y_{i_{1}}+Y_{i_{2}}\right) / 2$ on the $x_{i}$ ? Why or why not?

Shu Naito
Shu Naito
Numerade Educator
01:31

Problem 26

Suppose that $Z_{1}, Z_{2}, Z_{3}, Z_{4}$ are random variables with $\operatorname{Var}\left(Z_{i}\right)=1$ and $\operatorname{Cov}\left(Z_{i}, Z_{j}\right)=\rho$ for $i \neq j .$ Use the matrix techniques we developed in Section 14.4.1 to show that $Z_{1}+Z_{2}+Z_{3}+Z_{4}$ is uncorrelated with $Z_{1}+Z_{2}-Z_{3}-Z_{4}$

Hoan Nguyen
Hoan Nguyen
Numerade Educator
02:02

Problem 27

For the standard linear model of Section $14.4 .2,$ show that
$$
\sigma^{2} I=\Sigma_{i \dot{Y}}+\Sigma_{\dot{z} \varepsilon}
$$
Conclude that
$$
n \sigma^{2}=\sum_{i=1}^{n} \operatorname{Var}\left(\hat{Y}_{i}\right)+\sum_{i=1}^{n} \operatorname{Var}\left(\hat{e}_{i}\right)
$$

Nick Johnson
Nick Johnson
Numerade Educator
02:08

Problem 28

Suppose that $X_{1}, \ldots, X_{n}$ are independent with mean $\mu_{i}$ and common variance $\sigma^{2},$ Let $Y=\sum_{i=1}^{n} a_{i} X_{i}$
a. Let $Z=\sum_{i=1}^{n} b_{i} X_{i}$. Use Theorem $D$ of Section 14.4 .1 to find $\operatorname{Cov}(Y, Z)$
b. Use Theorem $\mathrm{C}$ of Section 14.4 .1 to find $E\left(\sum_{i=1}^{n} \sum_{j=1}^{n} X_{i} X_{j}\right)$

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 29

Assume that $X_{1}$ and $X_{2}$ are uncorrelated random variables with variance $\sigma^{2},$ and use matrix methods to show that $Y=X_{1}+X_{2}$ and $Z=X_{1}-X_{2}$ are uncorrelated. (Hint: Find $\Sigma_{\gamma z .}$ )

Rashmi Sinha
Rashmi Sinha
Numerade Educator
View

Problem 30

Let $X_{1}, \ldots, X_{n}$ be random variables with $\operatorname{Var}\left(X_{i}\right)=\sigma^{2}$ and $\operatorname{Cov}\left(X_{i}, X_{j}\right)=$
$\rho \sigma^{2},$ for $i \neq j .$ Use matrix methods to find $\operatorname{Var}(\bar{X})$

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:28

Problem 31

Let $Z$ be a random vector with 4 components and covariance matrix $\sigma^{2} I .$ Let $U=Z_{1}+Z_{2}+Z_{3}+Z_{4}$ and $V=\left(Z_{1}+Z_{2}\right)-\left(Z_{3}+Z_{4}\right) .$ Use matrix methods
to find $\operatorname{Cov}(U, V)$

Hoan Nguyen
Hoan Nguyen
Numerade Educator
View

Problem 32

Let $X$ be a random $n$ -vector and let $Y$ be a random vector with $Y_{1}=X_{1}$ $Y_{i}=X_{i}-X_{i-1}, i=1,2, \ldots, n$
a. If the $X_{i}$ are independent random variables with variances $\sigma^{2},$ find the covariance matrix of Y.
b. If the $Y_{i}$ are independent random variables with variances $\sigma^{2},$ find the covariance matrix of X.

Shu Naito
Shu Naito
Numerade Educator
View

Problem 33

a. Let $X \sim N(0,1)$ and $E \sim N(0,1)$ be independent, and let $Y=X+\beta E$ Show that
"
$$
r_{x y}=\frac{1}{\sqrt{\beta^{2}+1}}
$$
b. Use the results of part (a) to generate bivariate samples $\left(x_{i}, y_{i}\right)$ of size 20 with population correlation coefficients $-.9,-.5,0, .5,$ and $.9,$ and compute the sample correlation coefficients.
c. Have a partner generate scatterplots as in part (b) and then guess the correlation coefficients.

Shu Naito
Shu Naito
Numerade Educator
View

Problem 34

Generate a bivariate sample of size 50 as in Problem 33 with correlation coefficient $.8 .$ Find the estimated regression line and the residuals. Plot the residuals versus $X$ and the residuals versus $Y$. Explain the appearance of the plots.

Shu Naito
Shu Naito
Numerade Educator
01:37

Problem 35

An investigator wants to use multiple regression to predict a variable, $Y,$ from two other variables, $X_{1}$ and $X_{2}$. She proposes forming a new variable $X_{3}=X_{1}+X_{2}$ and using multiple regression to predict $Y$ from the three $X$ variables. Show that she will run into problems because the design matrix will not have full rank.

Hoan Nguyen
Hoan Nguyen
Numerade Educator
06:16

Problem 36

The file bi smuth contains the transition pressure (bar) of the bismuth II-I transition as a function of temperature $\left(^{\circ} \mathrm{C}\right)$ (see Example $\mathrm{E}$ in Section 14.2 .2 ). Fit a linear relationship between pressure and temperature, examine the residuals, and comment.

Banhishikha Sinha
Banhishikha Sinha
Numerade Educator
View

Problem 37

Dissociation pressure for a reaction involving barium nitride was recorded as a function of temperature (Orcutt 1970 ). The second law of thermodynamics gives the approximate relationship
$$
\ln (\text { pressure })=A+\frac{B}{T}
$$
where $T$ is absolute temperature. From the data in the file bar ium, estimate $A$ and $B$ and their standard errors. Form approximate $95 \%$ confidence intervals for
$A$ and $B$. Examine the residuals and comment.

Shu Naito
Shu Naito
Numerade Educator
View

Problem 38

The file sapphire lists observed values of Young's modulus ( $g$ ) measured at various temperatures $(T)$ for sapphire rods (Ku 1969). Fit a linear relationship $g=\beta_{0}+\beta_{1} t,$ and form confidence intervals for the coefficients. Examine the residuals.

Shu Naito
Shu Naito
Numerade Educator
View

Problem 39

As part of a nuclear safeguards program, the contents of a tank are routinely measured. The determination of volume is made indirectly by measuring the difference in pressure at the top and at the bottom of the tank. The tank is cylindrical
in shape, but its internal geometry is complicated by various pipes and agitator paddles. Without these complications, pressure and volume should have a linear relationship. To calibrate pressure with respect to volume, known quantities $(x)$ of liquid are placed in the tank and pressure readings $(y)$ are taken. The data in the file tankvolume are from Knafl et al. (1984). The units of volume are kiloliters and those of pressure are pascals.
a. Plot pressure versus volume. Does the relationship appear linear?
b. Calculate the linear regression of pressure on volume, and plot the residuals versus volume. What does the residual plot show?
c. Try fitting pressure as a quadratic function of volume. What do you think of the fit?

Shu Naito
Shu Naito
Numerade Educator
02:29

Problem 40

The following data come from the calibration of a proving ring, a device for measuring force (Hockersmith and Ku 1969).
a. Plot load versus deflection. Does the plot look linear?
b. Fit deflection as a linear function of load, and plot the residuals versus load. Do the residuals show any systematic lack of fit?
c. Fit deflection as a quadratic function of load, and estimate the coefficients and their standard errors. Plot the residuals. Does the fit look reasonable?
$$\begin{array}{cccc}
\hline \multicolumn{5}{c} {\text { Deflection }} \\
\hline \text { Load } & \text { Run 1 } & \text { Run 2 } & \text { Run 3 } \\
\hline 10,000 & 68.32 & 68.35 & 68.30 \\
20,000 & 136.78 & 136.68 & 136.80 \\
30,000 & 204.98 & 205.02 & 204.98 \\
40,000 & 273.85 & 273.85 & 273.80 \\
50,000 & 342.70 & 342.63 & 342.63 \\
60,000 & 411.30 & 411.35 & 411.28 \\
70,000 & 480.65 & 480.60 & 480.63 \\
80,000 & 549.85 & 549.85 & 549.83 \\
90,000 & 619.00 & 619.02 & 619.10 \\
100,000 & 688.70 & 688.62 & 688.58 \\
\hline
\end{array}$$

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 41

The file chestnut contains the diameter (feet) at breast height (DBH) and the age (years) of 27 chestnut trees (Chapman and Demeritt 1936 ). Try fitting DBH as a linear function of age. Examine the residuals. Can you find a transformation of DBH and/or age that produces a more linear relationship?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
View

Problem 42

The stopping distance $(y)$ of an automobile on a certain road was studied as a function of velocity (Brownlee 1960 ). The data are listed in the following table.
Fit $y$ and $\sqrt{y}$ as linear functions of velocity, and examine the residuals in each case. Which fit is better? Can you suggest any physical reason that explains why?
\begin{tabular}{cc}
\hline Velocity $(\mathrm{mi} / \mathrm{h})$ & Stopping Distance $(\mathrm{ft})$ \\
\hline 20.5 & 15.4 \\
20.5 & 13.3 \\
30.5 & 33.9 \\
40.5 & 73.1 \\
48.8 & 113.0 \\
57.8 & 142.6 \\
\hline
\end{tabular}

Shu Naito
Shu Naito
Numerade Educator
01:04

Problem 43

Chang (1945) studied the rate of sedimentation of amoebic cysts in water, in attempting to develop methods of water purification. The following table gives the diameters of the cysts and the times required for the cysts to settle through
$720 \mu \mathrm{m}$ of still water at three temperatures. Each entry of the table is an average of several observations, the number of which is given in parentheses. Does the time required appear to be a linear or a quadratic function of diameter? Can you find a model that fits? How do the settling rates at the three temperatures compare? (See Problem 7.)
$$\begin{array}{c|c|c|c}
\hline & \multicolumn{3}{|c} {\text { Setling Times of Cysts (sec) }} \\
\hline \text { Diameter }(\mu \mathrm{m}) & 10^{\circ} \mathrm{C} & 25^{\circ} \mathrm{C} & 28^{\circ} \mathrm{C} \\
\hline 11.5 & 217.1(2) & 138.2(1) & 128.4(2) \\
13.1 & 168.3(3) & 109.3(3) & 103.1(4) \\
14.4 & 136.6(11) & 89.1(13) & 82.7(11) \\
15.8 & 114.6(17) & 73.0(11) & 70.5(18) \\
17.3 & 96.4(8) & 61.3(6) & 59.7(6) \\
18.7 & 80.8(5) & 56.2(4) & 50.0(4) \\
20.2 & 70.4(2) & 46.3(1) & 41.4(2)
\end{array}$$

Nick Johnson
Nick Johnson
Numerade Educator
View

Problem 44

Cogswell (1973) studicd a method of measuring resistance to breathing in children. The file ast hma lists respiratory resistance and height (cm) for children with asthma and the file cyst fibr contains results for children with cystic fibrosis. Is there a statistically significant relation between respiratory resistance and height in either group?

Rashmi Sinha
Rashmi Sinha
Numerade Educator
View

Problem 45

The file reading contains average reading scores of third-graders from several elementary schools on a standardized test in each of two successive years. Is there
a "regression effect"?

Shu Naito
Shu Naito
Numerade Educator
View

Problem 46

Measurement of the concentration of small asbestos fibers is important in studies of environmental health issues and in setting and enforcing appropriate regulations. The concentrations of such fibers are measured most accurately by an electron microscope, but for practical reasons, optical microscopes must sometimes be used. Kiefer et al. (1987) compared measurements of asbestos fiber concentration from 30 airborne samples by a scanning electron microscope (SEM) and by a phase contrast microscope (PCM). The data are contained in the file asbestos. Study the relationship between the two measurements, taking the more accurate SEM measurements as the independent variable and the PCM measurements as the dependent variable.

Shu Naito
Shu Naito
Numerade Educator
17:40

Problem 47

Aerial survey methods are used to estimate the number of snow geese in their summer range areas west of Hudson's Bay in Canada. To obtain estimates, small aircraft fly over the range and, when a flock of snow geese is spotted, an experienced observer estimates the number of geese in the flock. To investigate the reliability of this method, an experiment in which an airplane carried two observers flew over 45 flocks, and each observer independently estimated the number of geese in the flock. Also, a photograph of the flock was taken so that an exact count of the number in the flock could be obtained (Weisberg 1985 ). The data are contained in the file geese.
a. Draw scatterplots of observer counts, $Y,$ versus photo count, $x .$ Do these graphs suggest that a simple linear regression model might be appropriate?
b. Calculate the linear regressions. What are the residual standard errors, what do they mean, and how do they compare? Do the fitted regressions appear to be different? Plot residuals and absolute values of residuals versus photo counts. Do the residuals indicate any systematic misfit? Does the residual variation appear to be constant?
c. Repeat the above using the square root transformation on the counts. Does this transformation stabilize the variance?
d. You have now computed the fits in two ways. How do they compare?
e. Write a few sentences in answer to the questions, "How well do observers estimate the number of geese?" "How do the two observers compare?"

Harsh Gadhiya
Harsh Gadhiya
Numerade Educator
01:52

Problem 48

The volume, height, and diameter at $4.5 \mathrm{ft}$ above ground level were measured for a sample of 31 black cherry trees in the Allegheny National Forest in Pennsylvania. The data were collected to provide a basis for determining an easy way of estimating the volume of a tree. Develop a model relating volume to height and diameter. The columns of the data matrix are diameter, height, and volume, in that order (Ryan, Joiner, \& Ryan 1976). The data are contained in the file treevolume.

Narayan Hari
Narayan Hari
Numerade Educator
02:15

Problem 49

The file flow-occ contains data collected by loop detectors in all three lanes (see Section 14.7). Examine the relationship of flow in lane 3 versus that in lane
1. Make a scatterplot and fit a regression line. Does the linear relationship look accurate or is there some systematic misfit? Fit local linear relationships with several bandwidths. Identify a bandwidth that is too small and one that is too large. What bandwidth appears to provide a good balance between being too wiggly and being over-smooth?

Danielle Nielsen
Danielle Nielsen
Numerade Educator
04:43

Problem 50

The file binary 59683 contains measurements of the light of an astronomical source as a function of time. Time is in units of days (Julian date), and brightness is measured as "magnitude." According to this system of measurement, the brightest star has magnitude -1.4 and the faintest visible star has magnitude 6 so the larger the magnitude, the dimmer the light.

Rakvi .
Rakvi .
Numerade Educator
03:45

Problem 51

The following table shows the monthly returns of stock in Disney. MacDonalds, Schlumberger, and Haliburton for January through May $1998 .$ Fit a multiple regression to predict Disney returns from those of the other stocks. What is the standard deviation of the residuals? What is $R^{2} ?$
$$\begin{array}{cccc}
\hline \text { Disney } & \text { MacDonalds } & \text { Schlumberger } & \text { Haliburton } \\
\hline 0.08088 & -0.01309 & -0.08463 & -0.13373 \\
0.04737 & 0.15958 & 0.02884 & 0.03616 \\
-0.04634 & 0.09966 & 0.00165 & 0.07919 \\
0.16834 & 0.03125 & 0.09571 & 0.09227 \\
-0.09082 & 0.06206 & -0.05723 & -0.13242 \\
\hline
\end{array}$$
Next, using the regression equation you have just found, carry out the predictions for January through May of 1999 and compare to the actual data listed below. What is the standard deviation of the prediction error? How can the comparison with the results from 1998 be explained? Is a reasonable explanation that the fundamental nature of the relationships changed in the one year period?
$$\begin{array}{cccc}
\hline \text { Disney } & \text { MacDonalds } & \text { Schlumberger } & \text { Haliburton } \\
\hline 0.1 & 0.02604 & 0.02695 & 0.00211 \\
0.06629 & 0.07851 & 0.02362 & -0.04 \\
-0.11545 & 0.06732 & 0.23938 & 0.35526 \\
0.02008 & -0.06483 & 0.06127 & 0.10714 \\
-0.08268 & -0.09029 & -0.05773 & -0.02933 \\
\hline
\end{array}$$

Sheryl Ezze
Sheryl Ezze
Numerade Educator
01:35

Problem 52

The file bodytemp contains normal body temperature readings (degrees Fahrenheit) and heart rates (beats per minute) of 65 males (coded by 1 ) and
65 females (coded by 2 ) from Shoemaker (1996).
a. For both males and females, make scatter plots of heart rate versus body temperature. Comment on the relationship or lack thereof.
b. Does the relationship for males appear to be the same as that for females? Examine this question graphically, by making a scatterplot showing both females and males and identifying females and males by different plotting symbols.
c. For the males, fit a linear regression to predict heart rate from temperature. Plot the residuals versus temperature and comment on whether the relationship is linear. Find the estimated slope and its standard error.
d. Repeat the above for females.
e. Test whether the slopes for males and females are equal. (Hint: Consider the difference of the slopes.)
f. Test whether the intercepts are equal.

Anand Jangid
Anand Jangid
Numerade Educator
05:31

Problem 53

Old Faithful geyser in Yellowstone National Park, Wyoming, derives its name from the regularity of its eruptions. The file oldfaithful contains measurements on eight successive days of the durations of the eruptions (in minutes) and the subsequent time interval before the next eruption. The park posts predicted eruption times for vistors. How well can the time until the next eruption be predicted by the duration of the current one?
a. Does the use of linear regression appear to be appropriate?
b. If the duration is 2 minutes, what would you predict the time until the next eruption to be? How can you quantify the accuracy of the prediction? Repeat this analysis for a duration of 4.5 minutes.

Julie Silva
Julie Silva
Numerade Educator
01:54

Problem 54

In $1970,$ Congress instituted a lottery for the military draft to support the unpopular war in Vietnam. All 366 possible birth dates were placed in plastic capsules in a rotating drum and were selected one by one. Eligible males born on the first day drawn were first in line to be drafted, etc. The results were criticized by some who claimed that government incompetency at running a fair lottery resulted in a tendency of men born later in the year being more likely to be drafted. Indeed, later investigation revealed that the birth dates were placed in the drum by month and were not thoroughly mixed. The columns of the file 1970lottery are month, month number, day of the year, and draft number.
a. Plot draft number versus day number. Do you see any trend?
b. Plot the linear regression line on the scatterplot.
c. Plot a local linear smoothing on the scatterplot. Try varying the bandwidth.

Neel Faucher
Neel Faucher
Numerade Educator
05:08

Problem 55

When gasoline is pumped into the tank of an automobile, hydrocarbon vapors in the tank are forced out and into the atmosphere, producing a significant amount of air pollution. For this reason, vapor-recovery devices are often installed on gasoline pumps. It is difficult to test a recovery device in actual operation, because all that can be measured is the amount of vapor actually recovered and, by means of a "sniffer," whether any vapor escaped into the atmosphere. To estimate the efficiency of the device, it is thus necessary to estimate the total amount of vapor in the tank by using its relation to the values of variables that can actually be measured. In this exercise, you will try to develop such a predictive relationship using data that were obtained in a laboratory experiment. The file gasvapor contains recordings of the following variables: initial tank temperature ( $\mathbf{F}$ ),temperature of the dispensed gasoline ( $\mathrm{F}$ ), initial vapor pressure in the tank (psi). vapor pressure of the dispensed gasoline (psi), and emitted hydrocarbons (g). A prediction of emitted hydrocarbons is desired.

First, randomly select 40 observations and set them aside. You will develop
a predictive relationship based on the remaining observations and then test its strength on the observations you have held out. (It is instructive to have each student in the class hold out the same 40 observations and then compare results.)
a. Look at the relationships among the variables by scatterplots. Comment on which relationships look strong. Based on this information, what variables would you conjecture will be important in the model? Do the plots suggest that transformations will be helpful? Do there appear to be any outliers?
b. Try fitting a few different models and select two that you think are the best.
c. Using these two models, predict the responses for the 40 observations you have held out and compare the predictions to the observed values by plotting predicted versus observed values, and by plotting prediction errors versus each of the independent variables. Summarize the strength of the prediction by the root mean square prediction error:
$$
\mathrm{RMSPE}=\sqrt{\frac{1}{40} \sum_{i=1}^{40}\left(Y_{i}-\hat{Y}_{i}\right)^{2}}
$$
where $Y_{i}$ is the $i$ th observed value and $\hat{Y}_{i}$ is the predicted value.

Adriano Chikande
Adriano Chikande
Numerade Educator
04:49

Problem 56

Recordings of the levels of pollutants and various meteorological conditions are made hourly at several stations by the Los Angeles Pollution Control District. This agency attempts to construct mathematical/statistical models to predict pollution levels and to gain a better understanding of the complexities of air pollution. Obviously, very large quantities of data are collected and analyzed, but only a small set of data will be considered in this problem. The file a irpollution contains the maximum level of an oxidant (a photochemical pollutant) and the morning averages of four meteorological variables: wind speed, temperature, humidity, and insolation (a measure of the amount of sunlight). The data cover 30 days during one summer.
a. Examine the relationship of oxidant level to each of the four meteorological variables and the relationships of the meteorological variables to each other. How well can the maximum level of oxidant be predicted from some or all of the meteorological variables? Which appear to be most important?
b. The standard statistical model used in this chapter assumes that the errors are random and independent of one another. In data that are collected over time, the error at any given time may well be correlated with the error from the preceding time. This phenomenon is called serial correlation, and in its presence the estimated standard errors of the coefficients developed in this chapter may be incorrect. The parameter estimates are still unbiased, however. (Why?) Can you detect serial correlation in the errors from your fits?

Robin Corrigan
Robin Corrigan
Numerade Educator