In a multiple regression analysis involving 4 predictor variables, and 25 observations, the total sum of squares is 800, and the error sum of squares is 200. The value of the F-test statistic for testing the usefulness of this model must be .

(A) 12
(B) 15
(C) 32
(D) 50
(E) 200

And then the F statistic would be given by:

And the correct answer would be:

(B) 15

Step-by-step explanation:

Previous concepts

Analysis of variance (ANOVA) "is used to analyze the differences among group means in a sample".  

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"  

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 independent variables and we have   individuals, we can define the following formulas of variation:  

And we have this property  

So then we have that:

The degrees of freedom for the model on this case is given by where k =4 represent the number of variables.

The degrees of freedom for the error on this case is given by . Since we know k we can find N.

And the total degrees of freedom would be

We can calculate the mean squares for the regression and the error like this:

And then the F statistic would be given by:

And the correct answer would be:

(B) 15

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