Question

1. Some data gathered by the Department of Education which relates education levels based on

scores: achievement tests given to high school students for example

urban: factor. Is the school located in an urban area?

distance: distance from a 4-year college (in 10 miles)

tuition: average state 4year college tuition (in 1000 USD).

        Coefficients:

               Estimate Std. Error t value Pr(>|t|)    

        (Intercept)  9.141015   0.148905  61.388  < 2e-16 ***

        score        0.095596   0.002679  35.686  < 2e-16 ***

        urbanyes     0.025619   0.057090   0.449   0.6536    

        distance    -0.048723   0.010539  -4.623 3.88e-06 ***

        tuition     -0.142627   0.068517  -2.082   0.0374 *  

        ---

        Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

        Residual standard error: 1.58 on 4734 degrees of freedom

        Multiple R-squared:  0.221,   Adjusted R-squared:  0.2203 

        F-statistic: 335.7 on 4 and 4734 DF, p-value: < 2.2e-16

1. What is the estimated regression equation?
2. What variables are significant?
3. How do you interpret the variable urban (yes/no)?
4. How much of the variance in the model can be explained by the variables that you included?
5. In general what information does VIF regression give you in assessing a model?
6. An output from R shown below, gives the Variance Inflation Factors for your model. What do these numbers tell you?

        score     urban     distance  tuition 

        1.031628  1.105871  1.112577  1.027281 

Answer 1

a.

The estimated regression equation is,

education levels = 9.141015 + 0.095596 Score + 0.025619 urbanyes - 0.048723 distance - 0.142627 tuition

b.

The variables are significant whihc have p-values below 0.05 and have * mentioned.

The significant variables are score, distance and tution.

c.

The difference in education level between schools located in urban and rural areas is 0.025619.

d.

The variance in the model explained by the included variables = Multiple R-squared = 0.221 = 22.1%

e.

A variance inflation factor(VIF) detects multicollinearity in regression analysis. Multicollinearity is when there’s correlation between predictors (i.e. independent variables) in a model. A high VIF can adversely affect your regression results.

f.

The numerical value for VIF tells you (in decimal form) what percentage the variance (i.e. the standard error squared) is inflated for each coefficient.

For Score, a VIF of 1.031628 tells you that the variance of a score coefficient is 3.16% bigger than what you would expect if there was no multicollinearity — if there was no correlation with other predictors.

For urban, a VIF of 1.105871 tells you that the variance of a urban coefficient is 10.59% bigger than what you would expect if there was no multicollinearity — if there was no correlation with other predictors.

For distance, a VIF of 1.112577 tells you that the variance of a distance coefficient is 11.26% bigger than what you would expect if there was no multicollinearity — if there was no correlation with other predictors.

For tution, a VIF of 1.027281 tells you that the variance of a tution coefficient is 2.73% bigger than what you would expect if there was no multicollinearity — if there was no correlation with other predictors.