Question

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

Answer 1

Answer:

$MS_{regression}= \frac{SS_{reg}}{k}= \frac{600}{4}=150$

$MS_{error}= \frac{SSE}{N-k-1}= \frac{200}{20}= 10$

And then the F statistic would be given by:

$F= \frac{MSR}{MSE}= \frac{150}{10}= 15$

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 $k$ independent variables and we have  $j=1,\dots,j$ individuals, we can define the following formulas of variation:  

$SS_{total}=\sum_{j=1}^n (y_j-\bar y)^2 = 800$  

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

$SS_{error}=\sum_{j=1}^n (y_{j}-\hat y_j)^2 =200$  

And we have this property  

$SST=SS_{regression}+SS_{error}$  

So then we have that:

$SST_{regression}= 800-200= 600$

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

The degrees of freedom for the error on this case is given by $df_{error}=N-k-1=25-4-1= 20$. Since we know k we can find N.

And the total degrees of freedom would be $df=N-1=25 -1 =24$

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

$MS_{regression}= \frac{SS_{reg}}{k}= \frac{600}{4}=150$

$MS_{error}= \frac{SSE}{N-k-1}= \frac{200}{20}= 10$

And then the F statistic would be given by:

$F= \frac{MSR}{MSE}= \frac{150}{10}= 15$

And the correct answer would be:

(B) 15