Question

In order to test for the significance of a regression model involving 14 independent variables and 255 observations, the numerator and denominator degree of freedom (respectively) for the critical value of F are Select one: a. 14 and 255. b. 255 and 14. c. 13 and 240. d. 14 and 240.

Answer 1

The degrees of freedom for the numerator on this case is given by
`df_(num)=k=14` where k =14 represent the number of groups.

The degrees of freedom for the denominator on this case is given by
`df_(den)=df_(between)=N-k-1=255-14-1=240`.

And the best option would be:

d. 14 and 240.

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"

If we assume that we have
`14` independent variables and on each group from
`j=1,\dots,k` we have
`k` individuals on each group we can define the following formulas of variation:

`SS_(total)=\sum_(j=1)^p \sum_(i=1)^(n_j) (x_(ij)-\bar x)^2`

`SS_(between=Treatment)=SS_(model)=\sum_(j=1)^p n_j (\bar x_(j)-\bar x)^2`

`SS_(within)=SS_(error)=\sum_(j=1)^p \sum_(i=1)^(n_j) (x_(ij)-\bar x_j)^2`

And we have this property

`SST=SS_(between)+SS_(within)`

The degrees of freedom for the numerator on this case is given by
`df_(num)=k=14` where k =14 represent the number of groups.

The degrees of freedom for the denominator on this case is given by
`df_(den)=df_(between)=N-k-1=255-14-1=240`.

And the best option would be:

d. 14 and 240.