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Chapter 12: Q143SE (page 807)

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=尾0+尾1x+尾2x2 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?

Short Answer

Expert verified

The researcher has fit a second-order model to the data and taken one measurement on y at 3 different values for x. The SSE value is coming out to be zero indicating that the data points lie perfectly on the line. However, since the SSE value is zero, the researcher might face issues while testing the significance of beta parameters or the goodness of fit test for the model.

Step by step solution

01

Second-order model

The researcher has fit a second-order model to the data and taken one measurement on y at 3 different values for x. The SSE value is coming out to be zero indicating that the data points lie perfectly on the line

02

Problems with second-order model

However, since the SSE value is zero, the researcher might face issues while testing the significance of beta parameters or the goodness of fit test for the model.

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Most popular questions from this chapter

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.

Question: Write a regression model relating E(y) to a qualitative independent variable that can assume three levels. Interpret all the terms in the model.

Question: The complete modelE(y)=尾0+尾1x1+尾2x2+尾3x3+尾4x4+蔚was fit to n = 20 data points, with SSE = 152.66. The reduced model,E(y)=尾0+尾1x1+尾2x2+蔚, was also fit, with

SSE = 160.44.

a. How many 尾 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 伪 = .05.

Suppose you fit the quadratic model E(y)=尾0+尾1x+尾2x2to a set of n = 20 data points and found R2=0.91, SSyy=29.94, and SSE = 2.63.

a. Is there sufficient evidence to indicate that the model contributes information for predicting y? Test using a = .05.

b. What null and alternative hypotheses would you test to determine whether upward curvature exists?

c. What null and alternative hypotheses would you test to determine whether downward curvature exists?
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