Question

If the coefficient of determination is 0.25 and the sum of squares residual is 180, then what is the value of SSY?

Answer 1

Answer:

And then $SSY=SS_{total}=\sum_{j=1}^n (y_j-\bar y)^2 =240$  

C. 240

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.

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:  

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

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

$SS_{error}=\sum_{j=1}^n (y_{j}-\hat y_j)^2 =180$  

And we have this property  

$SST==SSY=SS_{regression}+SS_{error}=SSR+180$

If we solve for SSR we got:

$SSR= SSY-180$    (1)

And we know that the determination coefficient is given by:

$R^2 = \frac{SSR}{SSY}$

We know the value os $R^2= 0.25$ and we can replace SSR in terms of SSY with the equation (1)  

$R^2 =0.25= \frac{SSY-180}{SSY}= 1-\frac{180}{SSY}$

And solving SSY we got:

$\frac{180}{SSY}=1-0.25=0.75$

$SSY= \frac{180}{0.75}=240$

And then $SSY=SS_{total}=\sum_{j=1}^n (y_j-\bar y)^2 =240$  

C. 240