Statistics / Statistics for Engineers and Scientists 4 / Chapter 8 / Problem 16SE

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

The article “Electrical Impedance Variation with Water

Chapter 8, Problem 16SE

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The article "Electrical Impedance Variation with Water Saturation in Rock" (Q. Su, Q. Feng, and Z. Shang, Geophysics, 2000:68-75) reports measurements of permeabilities (in $10^{-3}\ \mu \mathrm m^{3}$), porosities (in percent), and surface area per unit volume of pore space (in $10^{4}\ \mathrm {c m}^{-1}$) for several rock samples. The results are presented in the following table, denoting In Permeability by $y$ , porosity by $x_1$, and surface area per unit volume by $x_2$.

$y$	$x_1$	$x_2$	$y$	$x_1$	$x_2$
$-0.27$	$19.83$	$9.55$	$0.58$	$10.52$	$20.03$
$2.58$	$17.93$	$10.97$	$-0.56$	$18.92$	$13.10$
$3.18$	$21.27$	$31.02$	$-0.49$	$18.55$	$12.78$
$1.70$	$18.67$	$28.12$	$-0.01$	$13.72$	$40.28$
$-1.17$	$7.98$	$52.35$	$-1.71$	$9.12$	$53.67$
$-0.27$	$10.16$	$32.82$	$-0.12$	$14.39$	$26.75$
$-0.53$	$17.86$	$57.66$	$-0.92$	$11.38$	$75.62$
$-0.29$	$13.48$	$21.10$	$2.18$	$16.59$	$9.95$
$4.94$	$17.49$	$9.15$	$4.46$	$16.77$	$7.88$
$1.94$	$14.18$	$11.72$	$2.11$	$18.55$	$88.10$
$3.74$	$23.88$	$5.43$	$-0.04$	$18.02$	$10.95$

a. Fit the model $y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{1} x_{2}+\varepsilon$. Compute the analysis of variance table.

b. Fit the model $y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\varepsilon$. Compute the analysis of variance table.

c. Fit the model $y=\beta_{0}+\beta_{1} x_{1}+\varepsilon$. Compute the analysis of variance table.

d. Compute the $F$ statistics for comparing the models in parts (b) and (c) with the model in part (a). Which model do you prefer? Why?

Equation Transcription:

$1{0}^{-3}\ \mu{}{m}^{3}$

$1{0}^{4}\ c{m}^{-1}$

${x}_{1}$

${x}_{2}$

${x}_{1}$

${x}_{2}$

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The article “Electrical Impedance Variation with Water

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$y$	\(x_1\)	\(x_2\)	$y$	\(x_1\)	\(x_2\)
$-0.27$	$19.83$	$9.55$	$0.58$	$10.52$	$20.03$
$2.58$	$17.93$	$10.97$	$-0.56$	$18.92$	$13.10$
$3.18$	$21.27$	$31.02$	$-0.49$	$18.55$	$12.78$
$1.70$	$18.67$	$28.12$	$-0.01$	$13.72$	$40.28$
$-1.17$	$7.98$	$52.35$	$-1.71$	$9.12$	$53.67$
$-0.27$	$10.16$	$32.82$	$-0.12$	$14.39$	$26.75$
$-0.53$	$17.86$	$57.66$	$-0.92$	$11.38$	$75.62$
$-0.29$	$13.48$	$21.10$	$2.18$	$16.59$	$9.95$
$4.94$	$17.49$	$9.15$	$4.46$	$16.77$	$7.88$
$1.94$	$14.18$	$11.72$	$2.11$	$18.55$	$88.10$
$3.74$	$23.88$	$5.43$	$-0.04$	$18.02$	$10.95$

The article “Electrical Impedance Variation with Water

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