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Chapter 12: Q. 84E (page 764)

Question: Consider the model:

y=尾0+尾1x1+尾2x2+尾3x3+蔚

where x1 is a quantitative variable and x2 and x3 are dummy variables describing a qualitative variable at three levels using the coding scheme

role="math" localid="1649846492724" x2=1鈥�鈥�鈥�iflevel鈥�20鈥�鈥�otherwise鈥�鈥�鈥�鈥�鈥�鈥�鈥�x3=1鈥�鈥�鈥�if鈥�level鈥�30鈥�鈥�鈥�otherwise

The resulting least squares prediction equation is y^=44.8+2.2x1+9.4x2+15.6x3

a. What is the response line (equation) for E(y) when x2 = x3 = 0? When x2 = 1 and x3 = 0? When x2 = 0 and x3 = 1?

b. What is the least squares prediction equation associated with level 1? Level 2? Level 3? Plot these on the same graph.

Short Answer

Expert verified

a. The response lines for when x2 = x3 = 0 is y^=44.8+2.2x1. The response line when x2 = 1 and x3 = 0 is y^=54.2+2.2x1. The response line for when x2 = 0 and x3 = 1 is y^=60.4+2.2x1.

b. Graph

Step by step solution

01

Response lines

The response line for when x2 = x3 = 0 will be

y^=44.8+2.2x1+9.4(0)+15.6(0)y^=44.8+2.2x1

The response line for when x2 = 1 and x3 = 0 will be

y^=44.8+2.2x1+9.4(1)+15.6(0)y^=(44.8+9.4)+2.2x1y^=54.2+2.2x1

The response line for when x2 = 0 and x3 = 1 will be

y^=44.8+2.2x1+9.4(0)+15.6(1)y^=(44.8+15.6)+2.2x1y^=60.4+2.2x1

02

Graph

The response line for when x2 = x3 = 0 will be

y^=44.8+2.2x1+9.4(0)+15.6(0)y^=44.8+2.2x1

The response line for when x2 = 1 and x3 = 0 will be

role="math" localid="1649847635365" y^=44.8+2.2x1+9.4(1)+15.6(0)y^=(44.8+9.4)+2.2x1y^=54.2+2.2x1

The response line for when x2 = 0 and x3 = 1 will be

y^=44.8+2.2x1+9.4(0)+15.6(1)y^=(44.8+15.6)+2.2x1y^=60.4+2.2x1

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Most popular questions from this chapter

Question: Predicting elements in aluminum alloys. Aluminum scraps that are recycled into alloys are classified into three categories: soft-drink cans, pots and pans, and automobile crank chambers. A study of how these three materials affect the metal elements present in aluminum alloys was published in Advances in Applied Physics (Vol. 1, 2013). Data on 126 production runs at an aluminum plant were used to model the percentage (y) of various elements (e.g., silver, boron, iron) that make up the aluminum alloy. Three independent variables were used in the model: x1 = proportion of aluminum scraps from cans, x2 = proportion of aluminum scraps from pots/pans, and x3 = proportion of aluminum scraps from crank chambers. The first-order model, , was fit to the data for several elements. The estimates of the model parameters (p-values in parentheses) for silver and iron are shown in the accompanying table.

(A) Is the overall model statistically useful (at 伪 = .05) for predicting the percentage of silver in the alloy? If so, give a practical interpretation of R2.

(b)Is the overall model statistically useful (at a = .05) for predicting the percentage of iron in the alloy? If so, give a practical interpretation of R2.

(c)Based on the parameter estimates, sketch the relationship between percentage of silver (y) and proportion of aluminum scraps from cans (x1). Conduct a test to determine if this relationship is statistically significant at 伪 = .05.

(d)Based on the parameter estimates, sketch the relationship between percentage of iron (y) and proportion of aluminum scraps from cans (x1). Conduct a test to determine if this relationship is statistically significant at 伪 = .05.

Question: Determine which pairs of the following models are 鈥渘ested鈥� models. For each pair of nested models, identify the complete and reduced model.

a.E(y)=尾0+尾1x1+尾2x2b.E(y)=尾0+尾1x1c.E(y)=尾0+尾1x1+尾2x12d.E(y)=尾0+尾1x1+尾2x2+尾3x1x2e.E(y)=尾0+尾1x1+尾2x2+尾3x1x2+尾4x21+尾5x22

Question: Reality TV and cosmetic surgery. Refer to the Body Image: An International Journal of Research (March 2010) study of the impact of reality TV shows on a college student鈥檚 decision to undergo cosmetic surgery, Exercise 12.43 (p. 739). The data saved in the file were used to fit the interaction model, E(Y)=尾0+尾1x1+尾2x4+尾3x1x4, where y = desire to have cosmetic surgery (25-point scale),x1= {1 if male, 0 if female}, and x4= impression of reality TV (7-point scale). From the SPSS printout (p. 739), the estimated equation is:y^=11.78-1.97x1+0.58x4-0.55x1x4

a. Give an estimate of the change in desire (y) for every 1-point increase in impression of reality TV show (x4) for female students.

b. Repeat part a for male students.

Do blondes raise more funds? During fundraising, does the physical appearance of the solicitor impact the level of capital raised? An economist at the University of Nevada- Reno designed an experiment to answer this question and published the results in Economic Letters (Vol. 100, 2008). Each in a sample of 955 households was contacted by a female solicitor and asked to contribute to the Center for Natural Hazards Mitigation Research. The level of contribution (in dollars) was recorded as well as the hair color of the solicitor (blond Caucasian, brunette Caucasian, or minority female).

a) Consider a model for the mean level of contribution, E(y), that allows for different means depending on the hair color of the solicitor. Create the appropriate number of dummy variables for hair color. (Use minority female as the base level.)

b) Write the equation of the model, part a, incorporating the dummy variables.

c) In terms of the b鈥檚 in the model, what is the mean level of contribution for households contacted by a blond Caucasian solicitor?

d) In terms of the b鈥檚 in the model, what is the difference between the mean level of contribution for households contacted by a blond solicitor and those contacted by a minority female?

e) One theory posits that blond solicitors will achieve the highest mean contribution level, but that there will be no difference between the mean contribution levels attained by brunette Caucasian and minority females. If this theory is true, give the expected signs of the鈥檚 in the model.

f) The researcher found the b estimate for the dummy variable for blond Caucasian to be positive and significantly different from 0 (p-value < 0.01). The尾estimate for the dummy variable for brunette Caucasian was also positive, but not significantly different from 0 (p-value < 0.10). Do these results support the theory, part e?

Write a model that relates E(y) to two independent variables鈥攐ne 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.
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