Fortune magazine publishes an annual list of the 100 best companies to work for.
ID: 3320531 • Letter: F
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. Is the large company the baseline? Is the increase of one unit hourly rate an increase of $1000 in annual salary?
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 Regression file is as follows.
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 55.85709042Explanation / Answer
1. Interpretation:
If the average annual salary for hourly employees increases by 1000 unit, the average annual salary (in '000s) for salaried employees increases by 1.224 unit.
If the company is a midsize company, the average annual salary for salaried employees will increase by -3.208216286*1000 = -3208.216286 unit.
If the company is a small company, the average annual salary for salaried employees will increase by 34.40215452*1000 = 34402.15452 unit.
2. Interpretation of the coefficient of determination:
57.4907473% of variation in the average annual salary for salaried employees will be explained by the above regression model
3. Interpretation of the Multiple Correlation Coefficient :
Since r = 0.758226532, so there is a strong linear relationship between the independent variables and the dependent variable.
4. For all three independent variables ,
For testing the significance of Hourly ($1000s), P-value = 0 < 0.05
For testing the significance of Size-Midsize, P-value = 0.80155 > 0.05
and for testing the significance of Size-Small, P-value = 0.0028371 < 0.05
So at 5% level of significance, we can conclude that Hourly ($1000s) or the average annual salary for hourly employees and size-small are significant.
5. For the global overall test,
P-value = 0 < 0.05, so at 5% level of significance, we can conclude that the overall model is significant.