The following estimated regression model was developed relating yearly income (y in $1000s) of 30 individuals with their age (x1) and their gender (x2) (0 if male and 1 if female).

ŷ = 30 + 0.7x1 + 3x2

Also provided are SST = 1200 and SSE = 384.
The multiple coefficient of determination is _____.

a. .50
b. .42
c. .68
d. .32

The sum of squares "is the sum of the square of variation, where variation is defined as the spread between each individual value and the grand mean"

The correlation coefficient is a "statistical measure that calculates the strength of the relationship between the relative movements of two variables". It's denoted by r and its always between -1 and 1. And is defined as:

[tex]r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^2 -(\sum x)^2][n\sum y^2 -(\sum y)^2]}}[/tex]

When we conduct a multiple regression we want to know about the relationship between several independent or predictor variables and a dependent or criterion variable.

Solution to the problem

If we assume that we have [tex]k[/tex] independent variables and we have [tex]j=1,\dots,j[/tex] individuals, we can define the following formulas of variation:

[tex]SS_{total}=\sum_{j=1}^n (y_j-\bar y)^2[/tex]

[tex]SS_{regression}=SS_{model}=\sum_{j=1}^n (\hat y_{j}-\bar y)^2 [/tex]

[tex]SS_{error}=\sum_{j=1}^n (y_{j}-\hat y_j)^2 [/tex]

And we have this property

[tex]SST=SS_{regression}+SS_{error}[/tex]

We can find [tex]SS_{regression}= SST-SS_{error}=1200-384=816[/tex]

The degrees of freedom for the model on this case is given by [tex]df_{model}=df_{regression}=k=2[/tex] where k =2 represent the number of variables.

The degrees of freedom for the error on this case is given by [tex]df_{error}=N-k-1=30-2-1=27[/tex].

The determination coefficient when we conduct a multiple regression is defined as:

[tex]R^2 = \frac{SSE}{SST}=\frac{384}{1200}=0.32[/tex]

So the best option is b. .42