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Chapter 12: 86P (page 765)

Write a model that relates E(y) to two independent variables—one quantitative and one qualitative at four levels. Construct a model that allows the associated response curves to be second-order but does not allow for interaction between the two independent variables.

Short Answer

Expert verified

A second-order model with one quantitative variable and one qualitative variable with 4 levels can be written as Ey=β0+β1x1+β2x12+β3x2+β4x3+β5x4.

Step by step solution

01

Variable conditions

There are two independent variables: one quantitative variable (say x1) with a model in second-order and one qualitative variable with 4 levels (for k levels, (k-1) no of variables will be introduced in the model; namely x2,x3, and x4). There are no interactions observed between the two independent variables.

02

Model for E(y)

A second-order model with one quantitative variable and one qualitative variable with 4 levels can be written asEy=β0+β1x1+β2x12+β3x2+β4x3+β5x4

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Most popular questions from this chapter

Question: Refer to Exercise 12.82.

a. Write a complete second-order model that relates E(y) to the quantitative variable.

b. Add the main effect terms for the qualitative variable (at three levels) to the model of part a.

c. Add terms to the model of part b to allow for interaction between the quantitative and qualitative independent variables.

d. Under what circumstances will the response curves of the model have the same shape but different y-intercepts?

e. Under what circumstances will the response curves of the model be parallel lines?

f. Under what circumstances will the response curves of the model be identical?

Question: Women in top management. Refer to the Journal of Organizational Culture, Communications and Conflict (July 2007) study on women in upper management positions at U.S. firms, Exercise 11.73 (p. 679). Monthly data (n = 252 months) were collected for several variables in an attempt to model the number of females in managerial positions (y). The independent variables included the number of females with a college degree (x1), the number of female high school graduates with no college degree (x2), the number of males in managerial positions (x3), the number of males with a college degree (x4), and the number of male high school graduates with no college degree (x5). The correlations provided in Exercise 11.67 are given in each part. Determine which of the correlations results in a potential multicollinearity problem for the regression analysis.

  1. The correlation relating number of females in managerial positions and number of females with a college degree: r =0.983.

  2. The correlation relating number of females in managerial positions and number of female high school graduates with no college degree: r =0.074.

  3. The correlation relating number of males in managerial positions and number of males with a college degree: r =0.722.

  4. The correlation relating number of males in managerial positions and number of male high school graduates with no college degree: r =0.528.

Question: Suppose the mean value E(y) of a response y is related to the quantitative independent variables x1and x2

E(y)=2+x1-3x2-x1x2

a. Identify and interpret the slope forx2.

b. Plot the linear relationship between E(y) andx2forx1=0,1,2, where.

c. How would you interpret the estimated slopes?

d. Use the lines you plotted in part b to determine the changes in E(y) for each x1=0,1,2.

e. Use your graph from part b to determine how much E(y) changes when3⩽x1⩽5and1⩽x2⩽3.

Question: Adverse effects of hot-water runoff. The Environmental Protection Agency (EPA) wants to determine whether the hot-water runoff from a particular power plant located near a large gulf is having an adverse effect on the marine life in the area. The goal is to acquire a prediction equation for the number of marine animals located at certain designated areas, or stations, in the gulf. Based on past experience, the EPA considered the following environmental factors as predictors for the number of animals at a particular station:

X1 = Temperature of water (TEMP)

X2 = Salinity of water (SAL)

X3 = Dissolved oxygen content of water (DO)

X4 = Turbidity index, a measure of the turbidity of the water (TI)

x5 = Depth of the water at the station (ST_DEPTH)

x6 = Total weight of sea grasses in sampled area (TGRSWT)

As a preliminary step in the construction of this model, the EPA used a stepwise regression procedure to identify the most important of these six variables. A total of 716 samples were taken at different stations in the gulf, producing the SPSS printout shown below. (The response measured was y, the logarithm of the number of marine animals found in the sampled area.)

a. According to the SPSS printout, which of the six independent variables should be used in the model? (Use α = .10.)

b. Are we able to assume that the EPA has identified all the important independent variables for the prediction of y? Why?

c. Using the variables identified in part a, write the first-order model with interaction that may be used to predict y.

d. How would the EPA determine whether the model specified in part c is better than the first-order model?

e.Note the small value of R2. What action might the EPA take to improve the model?

Minitab was used to fit the complete second-order modeE(y)=β0+β1x1+β2x2+β3x1x2+β4x12+β5x22to n = 39 data points. The printout is shown on the next page.

a. Is there sufficient evidence to indicate that at least one of the parameters—β1,β2,β3,β4, andβ1,β2,β3,β4—is nonzero? Test usingα=0.05.

b. TestH0:β4=0againstHa:β4≠0. Useα=0.01.

c. TestH0:β5=0againstHa:β5≠0. Useα=0.01.

d. Use graphs to explain the consequences of the tests in parts b and c.
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