Problem 1. Regression Analysis In trying to find new locations for their restaurants, fast food chains such as McDonald's or Wendy's usually consider a number of factors. A company researcher is developing a model for site selection. Since high sales is the objective in site selection, the dependent variable in an analysis of sites will be annual gross sales. The independent variables will be mean annual household income (INC) and mean age of children (AGE) in the area surrounding the site. Twenty five currently operating sites with approximately 5,000 in population and one major competitor were randomly selected. A complete second-order model was used in the study. The model is of the form:

where:

y = annual gross sales

INC = mean annual household income

AGE = mean age of children in the area surrounding the site

The following least squares model is obtained:

with:

the standard error of b ₁ = 49.6

the standard error of b ₂ = 47.2

the standard error of b ₃ = 0.54

the standard error of b ₄ = 1.17

the standard error of b ₅ = 0.94

R ² = 0.91

F = 36.9 with 5 and 19 df.

1. Perform tests to determine if the quadratic terms are appropriate.
2. Is there significant interaction between income and mean age of children? Test an appropriate hypothesis.
3. At what level of income does sales reach an optimum (peak or trough) when mean age is 10 years?
4. An area under consideration for a restaurant has a median income of $35,000 and the average age of children is ten years, use this model to predict Sales in this area.

Problem 2. Violation of OLS conditions, time-series.

1. What the consequences of first order autocorrelation or errors (serial correlation) in the regression model?
2. A printout is supplied of a fit of the number of life insurance policies in force on the lives of U.S. residents for the years 1980 through 2006 using the model E(Y) = B0 + B1 YEAR. Examine the results presented here.
   Is there evidence of autocorrelation of error terms present in these results? Summarize this evidence and be sure to test an appropriate hypothesis for autocorrelation of errors.
3. Describe a correction procedure for the above problem if the first-order autocorrelation coefficient is known to be 0.75.

Problem 3

Nine similar machines are used in a manufacturing process at an assembly plant. The operations manager suspects that the repair costs for the machines are influenced by the operator of the machine. From past experience he knows that the age of the machine affects repair costs as well. The data on the next page shows repair costs (COST) for the machines over the past six months, the age (AGE) of each machine in years at the beginning of the six month period, the number of items processed (ITEMS) by the machine during the period, and the operator of the machine (OPER). Two dummy variables are also presented to capture the three levels of the variable OPER. They are listed as DA and DB in the table.

1. What does this model predict to be the effect of age of the machine on repair cost?
2. Does the age of the machine have a significant impact on repair costs of the machine as the manager suspected?
3. Report the estimated regression equation for operator B.
4. Interpret the numeric value of the coefficient for the variable DA.
5. Is there a significant difference in mean repair cost for operator B and operator C? Test an appropriate hypothesis.
6. Test both dummy variables, DA and DB, jointly for significance using an incremental sum of squares test.
7. Define multicollinearity and heteroscedasticity. Is there any evidence that either of these problems exist in this model?
8. Report the results of a formal test for non-normality of the error terms in this regression.
9. A second order model is presented on the last page of the printout. Is this second order model an improvement over the first order model regarding the objective of the study?

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Solution: The downloadable solution consists of 10 pages, 1305 words and 18 charts.
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