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Chapter 12: Q134SE (page 807)

Suppose you have developed a regression model to explain the relationship between y and x1, x2, and x3. The ranges of the variables you observed were as follows: 10 ≤ y ≤ 100, 5 ≤ x1 ≤ 55, 0.5 ≤ x2 ≤ 1, and 1,000 ≤ x3 ≤ 2,000. Will the error of prediction be smaller when you use the least squares equation to predict y when x1 = 30, x2 = 0.6, and x3 = 1,300, or when x1 = 60, x2 = 0.4, and x3 = 900? Why?

Short Answer

Expert verified

Therefore, when predicting y values, the error of prediction will be smaller when x1 = 30, x2 = 0.6, and x3 = 1300 since the values of independent variables are well within the range described in the question.

Step by step solution

01

Range of independent variables

The range of x1, x2, and x3is given as 5 ≤ x1≤ 55, 0.5 ≤ x2≤ 1, and 1,000 ≤ x3≤ 2,000. When x1= 30, x2= 0.6, and x3= 1300, all the variables x1,x2and x3are well within the range of values. While when x1= 60, x2= 0.4, and x3= 900, x1and x2are out of the range and x3is within the range.

02

Conclusion

Therefore, when predicting y values, the error of prediction will be smaller when x1 = 30, x2 = 0.6, and x3 = 1300.

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Most popular questions from this chapter

Question: Tipping behaviour in restaurants. Can food servers increase their tips by complimenting the customers they are waiting on? To answer this question, researchers collected data on the customer tipping behaviour for a sample of 348 dining parties and reported their findings in the Journal of Applied Social Psychology (Vol. 40, 2010). Tip size (y, measured as a percentage of the total food bill) was modelled as a function of size of the dining party(x1)and whether or not the server complimented the customers’ choice of menu items (x2). One theory states that the effect of the size of the dining party on tip size is independent of whether or not the server compliments the customers’ menu choices. A second theory hypothesizes that the effect of size of the dining party on tip size is greater when the server compliments the customers’ menu choices as opposed to when the server refrains from complimenting menu choices.

a. Write a model for E(y) as a function of x1 and x2 that corresponds to Theory 1.

b. Write a model for E(y) as a function of x1and x2that corresponds to Theory 2.

c. The researchers summarized the results of their analysis with the following graph. Based on the graph, which of the two models would you expect to fit the data better? Explain.

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’s decision to undergo cosmetic surgery, Exercise 12.17 (p. 725). Recall that the data for the study (simulated based on statistics reported in the journal article) are saved in the file. Consider the interaction model, , where y = desire to have cosmetic surgery (25-point scale), = {1 if male, 0 if female}, and = impression of reality TV (7-point scale). The model was fit to the data and the resulting SPSS printout appears below.

a.Give the least squares prediction equation.

b.Find the predicted level of desire (y) for a male college student with an impression-of-reality-TV-scale score of 5.

c.Conduct a test of overall model adequacy. Use a= 0.10.

d.Give a practical interpretation of R2a.

e.Give a practical interpretation of s.

f.Conduct a test (at a = 0.10) to determine if gender (x1) and impression of reality TV show (x4) interact in the prediction of level of desire for cosmetic surgery (y).

Consider fitting the multiple regression model

E(y)= β0+β1x1+ β2x2+β3x3+ β4x4 +β5x5

A matrix of correlations for all pairs of independent variables is given below. Do you detect a multicollinearity problem? Explain

To model the relationship between y, a dependent variable, and x, an independent variable, a researcher has taken one measurement on y at each of three different x-values. Drawing on his mathematical expertise, the researcher realizes that he can fit the second-order model Ey=β0+β1x+β2x2 and it will pass exactly through all three points, yielding SSE = 0. The researcher, delighted with the excellent fit of the model, eagerly sets out to use it to make inferences. What problems will he encounter in attempting to make inferences?

Question: The complete modelE(y)=β0+β1x1+β2x2+β3x3+β4x4+εwas fit to n = 20 data points, with SSE = 152.66. The reduced model,E(y)=β0+β1x1+β2x2+ε, was also fit, with

SSE = 160.44.

a. How many β parameters are in the complete model? The reduced model?

b. Specify the null and alternative hypotheses you would use to investigate whether the complete model contributes more information for the prediction of y than the reduced model.

c. Conduct the hypothesis test of part b. Use α = .05.
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