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Chapter 12: Q5E (page 722)

Question:How is the number of degrees of freedom available for estimating σ2(the variance ofε ) related to the number of independent variables in a regression model?

Short Answer

Expert verified

To estimate the value of the coefficients of any K+1 independent variable,K+1 numbers of equations are needed to mathematically solve it and find unique solutions to the equations

Step by step solution

01

Step-by-Step SolutionStep 1: No of independent variables in the model

To estimate the value of the coefficients of any K+1 independent variable,K+1numbers of equations are needed to mathematically solve it and find unique solutions to the equations

02

Degrees of freedom

Therefore, the no of degrees of freedom available for estimating the variance of σ2, would be n-(k+1).

Degree of freedom: degree of freedom is the number of values which are allowed to vary in the final calculation of a statistic.

In this case, the variance of error term is under discussion.

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Most popular questions from this chapter

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’s 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: Novelty of a vacation destination. Many tourists choose a vacation destination based on the newness or uniqueness (i.e., the novelty) of the itinerary. The relationship between novelty and vacationing golfers’ demographics was investigated in the Annals of Tourism Research (Vol. 29, 2002). Data were obtained from a mail survey of 393 golf vacationers to a large coastal resort in the south-eastern United States. Several measures of novelty level (on a numerical scale) were obtained for each vacationer, including “change from routine,” “thrill,” “boredom-alleviation,” and “surprise.” The researcher employed four independent variables in a regression model to predict each of the novelty measures. The independent variables were x1 = number of rounds of golf per year, x2 = total number of golf vacations taken, x3 = number of years played golf, and x4 = average golf score.

  1. Give the hypothesized equation of a first-order model for y = change from routine.
  1. A test of H0: β3 = 0 versus Ha: β3< 0 yielded a p-value of .005. Interpret this result if α = .01.
  1. The estimate of β3 was found to be negative. Based on this result (and the result of part b), the researcher concluded that “those who have played golf for more years are less apt to seek change from their normal routine in their golf vacations.” Do you agree with this statement? Explain.
  1. The regression results for three dependent novelty measures, based on data collected for n = 393 golf vacationers, are summarized in the table below. Give the null hypothesis for testing the overall adequacy of the first-order regression model.
  1. Give the rejection region for the test, part d, for α = .01.
  1. Use the test statistics reported in the table and the rejection region from part e to conduct the test for each of the dependent measures of novelty.
  1. Verify that the p-values reported in the table support your conclusions in part f.
  1. Interpret the values of R2 reported in the table.

Question: After-death album sales. When a popular music artist dies, sales of the artist’s albums often increase dramatically. A study of the effect of after-death publicity on album sales was published in Marketing Letters (March 2016). The following data were collected weekly for each of 446 albums of artists who died a natural death: album publicity (measured as the total number of printed articles in which the album was mentioned at least once during the week), artist death status (before or after death), and album sales (dollars). Suppose you want to use the data to model weekly album sales (y) as a function of album publicity and artist death status. Do you recommend using stepwise regression to find the “best” model for predicting y? Explain. If not, outline a strategy for finding the best model.

Question: There are six independent variables, x1, x2, x3, x4, x5, and x6, that might be useful in predicting a response y. A total of n = 50 observations is available, and it is decided to employ stepwise regression to help in selecting the independent variables that appear to be useful. The software fits all possible one-variable models of the form

where xi is the ith independent variable, i = 1, 2, …, 6. The information in the table is provided from the computer printout.

E(Y)=β0+β1xi

a. Which independent variable is declared the best one variable predictor of y? Explain.

b. Would this variable be included in the model at this stage? Explain.

c. Describe the next phase that a stepwise procedure would execute.

Production technologies, terroir, and quality of Bordeaux wine. In addition to state-of-the-art technologies, the production of quality wine is strongly influenced by the natural endowments of the grape-growing region—called the “terroir.” The Economic Journal (May 2008) published an empirical study of the factors that yield a quality Bordeaux wine. A quantitative measure of wine quality (y) was modeled as a function of several qualitative independent variables, including grape-picking method (manual or automated), soil type (clay, gravel, or sand), and slope orientation (east, south, west, southeast, or southwest).

  1. Create the appropriate dummy variables for each of the qualitative independent variables.
  2. Write a model for wine quality (y) as a function of grape-picking method. Interpret theβ’s in the model.
  3. Write a model for wine quality (y) as a function of soil type. Interpret theβ’s in the model.
  4. Write a model for wine quality (y) as a function of slope orientation. Interpret theβ’s in the model.
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