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.

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

1. Here the average annual salary for salaried employees is the dependent variable and the average annual salary for hourly employees and the size of the company.are the independent variables. So predicting the dependent variable basis the independent variable is a case of multiple linear regreassion.

Now the equation for this is Y=Bo + B1.X1 + B2.X2 + B3.X3 where X1 is Hourly wages, X1 is Size-Midsize and X2 is Size-Small and Y is salary for salaried employees

So here Y=26.96 + 1.22.X1 - 3.20 X2 + 34.4 X3 is the regression equation.

Here 26.96 is the y intercept of the line, i.e. the point at which the regression line or line of best fit crosses the y axis. The coefficients 1.22, -3.2, +34.4 are slopes or they descripe how y is related to these variables, so for every increase in Size Midsize, y will go down and will increase with the increase of the other two variables

2. The coeficient of determination tells us how fell the model or the regression equation fits the dataset.

Coeff of determination = R^2. Its value will lie between 0 and 1. The more closer it is 1, the better it fits the model, in this case it is 0.57 so this line explains about 57% of the variablityin the data

3. The multiple correlation coeff too tells how strongly the independent variable y is correlated to the dependent variables which is this case it 0.75, so y is strongly correlated to X1,X2 and X3 and hense these variables can be used to predict y

4. The t value is calcluated using the equation t= bi/sbi

So t for X1 = 1.22/0.25 = 4.88

x2 = -3.2/12.63 = -0.253

x3 = 34.4/10.43 = 3.29

So at a significance level of 0.05, X1 and X3 will be significant as the p values to the right if this t statistic would be less than 0.05, hence we will reject the null hyp that their coeff = 0.

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