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Chapter 12: 144SE (page 807)

Suppose you fit the regression model Ey=尾0+尾1x1+尾2x2+尾3x2+尾4x1x2+尾5x1x222 to n = 35 data points and wish to test the null hypothesisH0:尾4=尾5=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 伪 = .05.

Short Answer

Expert verified
  1. The alternate hypothesis to test the significance of interaction terms would be Ha: At least one of the parameters 尾4 or 尾5 is nonzero.
  2. The F-statistic to check the goodness of fit of the model can be computed byF test statistic =.SSEn-k+1.
  3. In part b, the degrees of freedom for numerator is (n-k) while the degree of freedom for denominator is [n-(k+1)].
  4. When 伪 = 0.05, the rejection region for the significance of interaction terms can be defined when the t-statistic < t0.025, n-1.

Step by step solution

01

Alternate hypothesis

The alternate hypothesis to test the significance of interaction terms would be Ha: At least one of the parameters 尾4 or 尾5 is nonzero.

02

F-statistic

The F-statistic to check the goodness of fit of the model can be computed by

F test statistic =SSEn-k+1 .

03

Degrees of freedom

In part b, the degrees of freedom for numerator is (n-k) while the degree of freedom for denominator is [n-(k+1)].

04

Rejection region

When 伪 = 0.05, the rejection region for the significance of interaction terms can be defined when the t-statistic < t0.025, n-1.

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Most popular questions from this chapter

Question: Chemical plant contamination. Refer to Exercise 12.18 (p. 725) and the U.S. Army Corps of Engineers study. You fit the first-order model,E(Y)=尾0+尾1x1+尾2x2+尾3x3 , to the data, where y = DDT level (parts per million),X1= number of miles upstream,X2= length (centimeters), andX3= weight (grams). Use the Excel/XLSTAT printout below to predict, with 90% confidence, the DDT level of a fish caught 300 miles upstream with a length of 40 centimeters and a weight of 1,000 grams. Interpret the result.

Cooling method for gas turbines. Refer to the Journal of Engineering for Gas Turbines and Power (January 2005) study of a high-pressure inlet fogging method for a gas turbine engine, Exercise 12.19 (p. 726). Recall that you fit a first-order model for heat rate (y) as a function of speed (x1) , inlet temperature (x2) , exhaust temperature (x3) , cycle pressure ratio (x4) , and airflow rate (x5) . A Minitab printout with both a 95% confidence interval for E(y) and prediction interval for y for selected values of the x鈥檚 is shown below.

a. Interpret the 95% prediction interval for y in the words of the problem.

b. Interpret the 95% confidence interval forE(y)in the words of the problem.

c. Will the confidence interval for E(y) always be narrower than the prediction interval for y? Explain.

Question: Ambiance of 5-star hotels. Although invisible and intangible, ambient conditions such as air quality , temperature , odor/aroma , music , noise level , and overall image may affect guests鈥� satisfaction with their stay at a hotel. A study in the Journal of Hospitality Marketing & Management (Vol. 24, 2015) was designed to assess the effect of each of these ambient factors on customer satisfaction with the hotel . Using a survey, researchers collected data for a sample of 422 guests at 5-star hotels. All variables were measured as an average of several 5-point questionnaire responses. The results of the multiple regression are summarized in the table on the next page.

  1. Write the equation of a first-order model for hotel image as a function of the six ambient conditions.
  2. Give a practical interpretation of each of the b-estimates shown.
  3. A 99% confidence interval for is (.350, .576). Give a practical interpretation of this result.
  4. Interpret the value of adjusted .
  5. Is there sufficient evidence that the overall model is statistically useful for predicting hotel image ? Test using a = .01.

Write a model relating E(y) to one qualitative independent variable that is at four levels. Define all the terms in your model.

Question: Risk management performance. An article in the International Journal of Production Economics (Vol. 171, 2016) investigated the factors associated with a firm鈥檚 supply chain risk management performance (y). Five potential independent variables (all measured quantitatively) were considered: (1) firm size, (2) supplier orientation, (3) supplier dependency, (4) customer orientation, and (5) systemic purchasing. Consider running a stepwise regression to find the best subset of predictors for risk management performance.

a. How many 1-variable models are fit in step 1 of the stepwise regression?

b. Assume supplier orientation is selected in step 1. How many 2-variable models are fit in step 2 of the stepwise regression?

c. Assume systemic purchasing is selected in step 2. How many 3-variable models are fit in step 3 of the stepwise regression?

d. Assume customer orientation is selected in step 3. How many 4-variable models are fit in step 4 of the stepwise regression?

e. Through the first 4 steps of the stepwise regression, determine the total number of t-tests performed. Assuming each test uses an a = .05 level of significance, give an estimate of the probability of at least one Type I error in the stepwise regression.
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