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Chapter 12: 142SE (page 708)

Consider relating E(y) to two quantitative independent variables x1 and x2.

  1. Write a first-order model for E(y).
  2. Write a complete second-order model for E(y).

Short Answer

Expert verified
  1. A first-order model for E(y) with two quantitative variables x1and x2 can be written asEy=尾0+尾1x1+尾2x2+蔚.
  2. A Second-order model for E(y) with two quantitative variables x1and x2 can be written asEy=尾0+尾1x1+尾2x2+尾3x12+尾4x22+蔚 .

Step by step solution

01

First-order model for E(y)

A first-order model for E(y) with two quantitative variables x1and x2 can be written as Ey=尾0+尾1x1+尾2x2+蔚.

02

Second-order model for E(y)

A Second-order model for E(y) with two quantitative variables x1and x2 can be written as Ey=尾0+尾1x1+尾2x2+尾3x12+尾4x22+蔚.

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Most popular questions from this chapter

Question: Tilting in online poker. In poker, making bad decisions due to negative emotions is known as tilting. A study in the Journal of Gambling Studies (March, 2014) investigated the factors that affect the severity of tilting for online poker players. A survey of 214 online poker players produced data on the dependent variable, severity of tilting (y), measured on a 30-point scale (where higher values indicate a higher severity of tilting). Two independent variables measured were poker experience (x1, measured on a 30-point scale) and perceived effect of experience on tilting (x2, measured on a 28-point scale). The researchers fit the interaction model, . The results are shown below (p-values in parentheses).

  1. Evaluate the overall adequacy of the model using 伪 = .01.

b. The researchers hypothesize that the rate of change of severity of tilting (y) with perceived effect of experience on tilting (x2) depends on poker experience (x1). Do you agree? Test using 伪 = .01.

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: 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 one鈥檚 desire to undergo cosmetic surgery, Exercise 12.17 (p. 725). Recall that psychologists used multiple regression to model desire to have cosmetic surgery (y) as a function of gender(x1) , self-esteem(x2) , body satisfaction(x3) , and impression of reality TV (x4). The SPSS printout below shows a confidence interval for E(y) for each of the first five students in the study.

  1. Interpret the confidence interval for E(y) for student 1.
  2. Interpret the confidence interval for E(y) for student 4

Question:Suppose you fit the first-order model y=尾0+尾1x1+尾2x2+尾3x3+尾4x4+尾5x5+蔚to n=30 data points and obtain SSE = 0.33 and R2=0.92

(A) Do the values of SSE and R2suggest that the model provides a good fit to the data? Explain.

(B) Is the model of any use in predicting Y ? Test the null hypothesis H0:尾1=尾2=尾3=尾4=尾5=0 against the alternative hypothesis {H}at least one of the parameters 尾1,尾2,...,尾5 is non zero.Use伪=0.05 .

Suppose you used Minitab to fit the model y=尾0+尾1x1+尾2x2+蔚

to n = 15 data points and obtained the printout shown below.

  1. What is the least squares prediction equation?

  2. Find R2and interpret its value.

  3. Is there sufficient evidence to indicate that the model is useful for predicting y? Conduct an F-test using 伪 = .05.

  4. Test the null hypothesis H0: 尾1= 0 against the alternative hypothesis Ha: 尾1鈮� 0. Test using 伪 = .05. Draw the appropriate conclusions.

  5. Find the standard deviation of the regression model and interpret it.
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