Question

A researcher evaluates the significance of a multiple-regression equation and obtains an F-ratio with df = 2,24. How many participants were in the sample?

Answer 1

N=27 participants

Explanation:

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`

And we have this property

`SST=SS_(regression)+SS_(error)`

The degrees of freedom for the model on this case is given by
`df_(model)=df_(regression)=k=2` where k =2 represent the number of variables.

The degrees of freedom for the error on this case is given by
`df_(error)=N-k-1=24`. Sinc we know k we can find N.

`N=24+k+1=24+2+1=27`

And the total degrees of freedom would be
`df=N-1=27 -1 =26`

On this case the correct answer would be N=27 participants