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 median 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:
E(y) = β0 + β1 INC + β2 AGE + β3 INC2 + β4 AGE2 + β5 INC*AGE
where:y = annual gross sales
INC = median annual household income
AGE = mean age of children in the area surrounding the site
The following least squares model is obtained:
Å¶ = -6055.5 + 322.2 INC – 15.8 AGE – 3.72 INC2 – 3.9 AGE2 + 2.0 INC*AGE
with: the standard error of b1 = 49.6 the standard error of b2 = 47.2 the standard error of b3 = 0.54 the standard error of b4 = 1.17 the standard error of b5 = 0.94
R2 = 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 and, if so, at what significance level? Test an appropriate hypothesis.
3.Based on this information, what effect does the mean age of children have on gross sales?
Hint: Your response will be a function of median income.
4.At what level of income does sales reach an optimum (peak or trough) when mean age is
5.An area under consideration for a restaurant has a median income is $45,000 and the average age of children is eight years, use this model to predict Sales in this area.
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? Test an appropriate hypothesis.
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 (Chow) 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? Are any of the second order terms in the model significant at the 5% level?
10.What overall conclusions can you make from this analysis?
|Due By (Pacific Time)||12/10/2014 10:00 am|
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