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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=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?

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

Suppose you fit the regression model Ey=β0+β1x1+β2x2+β3x22+β4x1x2+β5x1x222 to n = 35 data points and wish to test the null hypothesis H0:β4=β5=0

  1. State the alternative hypothesis.

  2. Explain in detail how to compute the F-statistic needed to test the null hypothesis.

  3. What are the numerator and denominator degrees of freedom associated with the F-statistic in part b?

  4. Give the rejection region for the test if α = .05.

Question: Bus Rapid Transit study. Bus Rapid Transit (BRT) is a rapidly growing trend in the provision of public transportation in America. The Center for Urban Transportation Research (CUTR) at the University of South Florida conducted a survey of BRT customers in Miami (Transportation Research Board Annual Meeting, January 2003). Data on the following variables (all measured on a 5-point scale, where 1 = very unsatisfied and 5 = very satisfied) were collected for a sample of over 500 bus riders: overall satisfaction with BRT (y), safety on bus (x1), seat availability (x2), dependability (x3), travel time (x4), cost (x5), information/maps (x6), convenience of routes (x7), traffic signals (x8), safety at bus stops (x9), hours of service (x10), and frequency of service (x11). CUTR analysts used stepwise regression to model overall satisfaction (y).

a. How many models are fit at step 1 of the stepwise regression?

b. How many models are fit at step 2 of the stepwise regression?

c. How many models are fit at step 11 of the stepwise regression?

d. The stepwise regression selected the following eight variables to include in the model (in order of selection): x11, x4, x2, x7, x10, x1, x9, and x3. Write the equation for E(y) that results from stepwise regression.

e. The model, part d, resulted in R2 = 0.677. Interpret this value.

f. Explain why the CUTR analysts should be cautious in concluding that the best model for E(y) has been found.

Question: Identify the problem(s) in each of the residual plots shown below.

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.

Consider a multiple regression model for a response y, with one quantitative independent variable x1 and one qualitative variable at three levels.

a. Write a first-order model that relates the mean response E(y) to the quantitative independent variable.

b. Add the main effect terms for the qualitative independent variable to the model of part a. Specify the coding scheme you use.

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 lines of the model in part c be parallel?

e. Under what circumstances will the model in part c have only one response line?
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