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Business / Statistics for Business and Economics 12 / Chapter 12 / Problem 47E

Statistics for Business and Economics | 12th Edition | ISBN: 9780321826237 | Authors: James T. McClave, P. George Benson, Terry T Sincich

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Extracting water from oil. Refer to the Journal of

Chapter 12, Problem 47E

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QUESTION:

Extracting water from oil. Refer to the Journal of Colloidand Interface Science study of water/oil mixtures, Exercise12.20 (p. 684). Recall that three of the seven variables used to predict voltage (y) were volume (\(x_1\)), salinity (\(x_2\)), and surfactant concentration (\(x_5\)). The model the researchers fit is

\(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{5}+\beta_{4} x_{1} x_{2}+\beta_{5} x_{1} x_{5}\)

a. Note that the model includes interaction between disperse phase volume (\(x_1\)) and salinity (\(x_2\)) as well as interaction between disperse phase volume (\(x_1\)) and surfactant concentration (\(x_5\)). Discuss how these interaction terms affect the hypothetical relationship between y and \(x_1\). Draw a sketch to support your answer.

b. Fit the interaction model to the data. Does this model appear to fit the data better than the first-order modeling Exercise 12.20? Explain.

c. Interpret the \(\beta\) estimates of the interaction model.

Questions & Answers

QUESTION:

Extracting water from oil. Refer to the Journal of Colloidand Interface Science study of water/oil mixtures, Exercise12.20 (p. 684). Recall that three of the seven variables used to predict voltage (y) were volume (\(x_1\)), salinity (\(x_2\)), and surfactant concentration (\(x_5\)). The model the researchers fit is

\(E(y)=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{3} x_{5}+\beta_{4} x_{1} x_{2}+\beta_{5} x_{1} x_{5}\)

a. Note that the model includes interaction between disperse phase volume (\(x_1\)) and salinity (\(x_2\)) as well as interaction between disperse phase volume (\(x_1\)) and surfactant concentration (\(x_5\)). Discuss how these interaction terms affect the hypothetical relationship between y and \(x_1\). Draw a sketch to support your answer.

b. Fit the interaction model to the data. Does this model appear to fit the data better than the first-order modeling Exercise 12.20? Explain.

c. Interpret the \(\beta\) estimates of the interaction model.

ANSWER:

Step 1 of 11

A first-order model for mean voltage (y)as a function of disperse phase volume (),

salinity () , temperature () , time delay () , surfactant concentration () , span: triton () and solid particles () is

Where are all quantitative variables that are not functions of other

independent variables and (i =1,2,3,4,5,6,7) represent the slope of the line relating

y to when all the other x values are held ?xed.

Given information is that three of the seven variables used to predict voltage ( y)were

volume() , salinity () , and surfactant concentration ().

The researcher model ?t is

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