A Multiple-regression equation with two predictor variables accounts for R ² = 2

Question

A Multiple-regression equation with two predictor variables accounts for R²= 28.5% of the variance in the Y scores for a sample of n=25 individuals.
a. If SSy=10, does the regression equation predict a significant proportion of the variance in the Y scores? Use a =.05 to evaluate the F-ratio. Be sure to show all formulas with symbols (and plug in numbers), steps, processes and calculations for all parts of the answers.
b. If SSy=100, does the regression equation predict a significant proportion of the variance in the Y scores? Use a =.05 to evaluate the F-ratio. Be sure to show all formulas with symbols (and plug in numbers), steps, processes and calculations for all parts of the answers.

Explanation / Answer

a) Given R^2 = 0.285
i.e. RSS / TSS = 0.285 where TSS is total sum of square or total variation
RSS = 0.285 * TSS
If SSy = 10 = TSS then RSS = 0.285*10 = 2.85
Error sum of square ESS = TSS - RSS = 10-2.85 = 7.15
Error mean sum of square EMSS = ESS / (n-k-1) = 7.15 / (25-2-1) = 0.325 where k is number of independent variables
Regression mean sum of square = RMSS = RSS / k =2.85/2 = 1.425
F-Ratio = RMSS / EMSS = 1.425/0.325= 4.3846
P-value = 0.024967006
Here P-value < alpha 0.05, so we reject H0
Thus we conclude that the regression equation is best fit to the given data

b) Given R^2 = 0.285
i.e. RSS / TSS = 0.285 where TSS is total sum of square or total variation
RSS = 0.285 * TSS
If SSy = 100 = TSS then RSS = 0.285*10 = 28.5
Error sum of square ESS = TSS - RSS = 100-28.5 = 71.5
Error mean sum of square EMSS = ESS / (n-k-1) = 71.5 / (100-2-1) = 0.737 where k is number of independent variables
Regression mean sum of square = RMSS = RSS / k = 28.5/2 = 14.25
F-Ratio = RMSS / EMSS = 14.25/0.737 = 19.333
P-value = 0.00000
Here P-value < alpha 0.05, so we reject H0
Thus we conclude that the regression equation is best fit to the given data

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