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Regression analysis

See attached file for full problem description.

Suppose that the sales manager of a company wishes to evaluate the performance of the company's sales representatives. Each sales representative is solely responsible for one sales territory, and the manager decides that it is reasonable to measure performance of a sales representative by using the yearly sales of the company's product in the representative's sales territory (Y). The manager feels that the sales performance depends on the following five independent variables:

X1 = number of months the representative has been employed by the company
X2 = sales of the company's product and competing products in the sales territory
X3 = dollar advertising expenditure in the territory
X4 = weighted average if the company's market share in the territory for the previous four years
X5 = change in the company's market share in the territory over the previous four years

a) Please write down the regression equation between Y as the dependent variable and X1,X2,X3,X4 and X5 as independent variables.

b) Please interpret all regression coefficients in the context of the above problem (i.e. β0, β1, ... ,β5).

c) Determine the standard error of the estimate, coefficient of determination and the adjusted coefficient of determination.

d) At the 0.01 level of significance, which of the independent variables have significant effects?

e) Using a 0.01 level of significance, do you think that at least one of the independent variables provide any explanatory power in predicting Y?

Please state your hypothesis clearly (both verbally and symbolically). Also briefly explain the testing procedure (namely what type of test you are using and why you are using it).

f) Calculate the VIFs for the variables X1 and X2.

g) Do you think there exists multi-colinearity in the model? Why or why not?

h-i-j) Please check the validity of your model, in other words check whether the
regression assumptions (all three of them) hold in your model. In doing so, please briefly
explain your reasoning (state your hypothesis', test statistics, rejection regions and
conclusions for all 3).

Please use the following JMP-IN output to in order to answer the above questions.

For the regression between Y and X1,...,X5
Summary of Fit
RSquare ?
RSquare Adj 0.892643
Root Mean Square Error ?
Mean of Response 3374.568
Observations (or Sum Wgts) 25

Analysis of Variance
Source DF Sum of Squares Mean Square F Ratio
Model 5 37862659 7572532 ?
Error 19 3516890 185099 Prob > F
C. Total 24 41379549 ?

Parameter Estimates
Term Estimate Std Error t Ratio Prob>|t|
Intercept -1113.788 419.8869 ? ?
TimewithCompany, X1 3.6121012 1.1817 ? ?
MarketPot, X2 0.0420881 0.006731 ? ?
Advertising, X3 0.1288568 0.037036 ? ?
Market Share, X4 256.95554 39.13607 ? ?
MarketShareChange, X5 324.53345 157.2831 ? ?

Durbin-Watson Number of Obs. AutoCorrelation
1.761867 25 0.1014

The distribution of the residuals from the regression of Y vs. X1,...,X5

Goodness-of-Fit Test
Shapiro-Wilk W Test
W Prob<W
0.944057 0.1836

For the regression between the squared residuals and X1, ..., X5
Summary of Fit
RSquare 0.235809
RSquare Adj 0.034707
Root Mean Square Error 165894.6
Mean of Response 140675.6
Observations (or Sum Wgts) 25

Parameter Estimates
Term Estimate Std Error t Ratio Prob>|t|
Intercept -205532 161905.7 -1.27 0.2196
TimewithCompany -249.9095 455.6559 -0.55 0.5898
MarketPot 4.3155604 2.595516 1.66 0.1128
Advertising 9.7092021 14.28088 0.68 0.5048
Market Share 22038.486 15090.62 1.46 0.1605
MarketShareChange -90037.44 60647.34 -1.48 0.1541

For the regression between X1 and X2,...,X5
Summary of Fit
RSquare 0.266888
RSquare Adj 0.120265
Root Mean Square Error 81.41049
Mean of Response 87.642
Observations (or Sum Wgts) 25

For the regression between X2 and X1,X3,X4,X5
Summary of Fit
RSquare 0.310709
RSquare Adj 0.172851
Root Mean Square Error 14292.02
Mean of Response 38858.05
Observations (or Sum Wgts) 25

Correlations between Y,X1,...,X5
Sales TimewithCompany MarketPot Advertising Market Share MarketShareChange
Sales 1.0000 0.6229 0.5978 0.5962 0.4835 0.4892
TimewithCompany 0.6229 1.0000 0.4540 0.2492 0.1062 0.2515
MarketPot 0.5978 0.4540 1.0000 0.1741 -0.2107 0.2683
Advertising 0.5962 0.2492 0.1741 1.0000 0.2645 0.3765
Market Share 0.4835 0.1062 -0.2107 0.2645 1.0000 0.0855
MarketShareChange 0.4892 0.2515 0.2683 0.3765 0.0855 1.0000

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Solution Summary

Step by step method for regression analysis is discussed here. Regression coefficients, coefficient of determination, scatter diagram and significance of regression model are explained in the solution.