Fortune magazine publishes an annual list of the 100 best companies to work for.

Question

Fortune magazine publishes an annual list of the 100 best companies to work for. The data in the file named FortuneBest shows a portion of the data for a random sample of 30 of the companies that made the top 100 list for 2012 (Fortune, February 6, 2012). The column labeled Rank shows the rank of the company in the Fortune 100 list; the column labeled Size indicates whether the company is a small, midsize, or large company; the column labeled Salaried ($1000s) shows the average annual salary for salaried employees rounded to the nearest$1000 ; and the column labeled Hourly ($1000s) shows the average annual salary for hourly employees rounded to the nearest $1000 . Fortune defines large companies as having more than 10,000employees, midsize companies as having between 2500 and 10,000 employees, and small companies as having fewer than 2500

To incorporate the effect of size, a categorical variable with three levels, we used two dummy variables: Size-Midsize and Size-Small. The value of size-Midsize =1   if the company is a midsize company and 0 otherwise. And, the value of size-small =1 if the company is a small company and 0 otherwise. Develop an estimated regression equation that could be used to predict the average annual salary for salaried employees given the average annual salary for hourly employees and the size of the company.

1. Interpret the regression constant and regression coefficients.

2. Interpret the coefficient of determination

3. Interpret the Multiple Correlation Coefficient

4. For the estimated regression equation developed above, use the t test to determine the significance of the independent variables. Use Alpha =0.05

5. Do a global overall test.

The Summary Output of the data file is as follows.

SUMMARY OUTPUT Regression Statistics Multiple R 0.758226532 R Square 0.574907473 Adjusted R Square 0.525858336 Standard Error 25.47515084 Observations 30 ANOVA df SS MS F Significance F Regression 3 22820.30059 7606.766865 11.72105159 4.81713E-05 Residual 26 16873.56607 648.9833105 Total 29 39693.86667 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 26.96589812 14.00431214 1.925542494 0.065164701 -1.820377748 55.75217398 -1.820377748 55.75217398 Hourly ($1000s) 1.224044868 0.258106065 4.742410332 6.63374E-05 0.693500254 1.754589482 0.693500254 1.754589482 Size-Midsize -3.208216286 12.63462389 -0.253922579 0.801552712 -29.17905763 22.76262506 -29.17905763 22.76262506 Size-Small 34.40215452 10.437669 3.295961437 0.0028371 12.94721862 55.85709042 12.94721862 55.85709042

Explanation / Answer

From the given output

1)  The intercept is the expected mean value of Y when all X=0. Here, it means that when all other indepdent variables are zero, then salary of the employees on average is 26.965 units

The coefficients:

Hourly coefficient of 1.22 tells that if average hourly salary increases by 1 unit then the total annual salary increases by 1.22 units

If the size of company is mid size, then salary decreases by 3.2 units per 1 unit increase of the size of company

If the size of company is small size, then salary increases by 34.4 units per 1 unit increase of the size of company

Also,

2) R2 = 0.5749 = 57.49%

i.e., 57.49% of variation in the annual salary was explained by the only the significant independent predictors( hourly and size-small)

3) R = 0.7582

A high positive correlation between all the variables. Here the coefficient of multiple correlation is a measure of how well annual salary can be predicted using a linear function of hourly salary and company size. It is the correlation between the variable's values and the best predictions that can be computed linearly from the predictive variables.

4) From the above table only two independent variables were found to be significant which was tested by student 't' statistic at 5% level of significance.

a) hourly (p=0.000<0.05)

and Size_small (p=0.0028<0.05)

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