# ANOVA determining percentage of variation

The Financial Review wished to estimate the amount of annual government spending using tax revenue and level of nationwide debt. Data from 1958-2008 (inclusive) was used. All variables were measured in billions of dollars. It was found that the mean square error for the regression was 15 and the total sum of squares was 12200. Hence, what percentage of the variation in annual government spending is explained by the regression equation? Give your answer correct to two decimal places.

My Perspective:

Mean square of error (MSE) = 15
Total sum of squares (SST) = 12200


The question is asking for R squared, so

$$R^2 = SSR / SST$$, but so far we only have $$SST$$. How do we derive $$SSR$$ from the $$MSE$$ in this case?

Think about how many degrees of freedom there must be (total, residual & regression). Then you can work backwards from the $MSE$ to get the missing $SS$s.

• df in this case is unknown because we do not know how many variables there are? I believe for MSE, it is SSE/(n-k-1), where k is the number of independent variables and n is the total number of variables. n is unknown from this question. May 11, 2013 at 15:44
• If it were true that the df is unknowable, then the question is unanswerable. Since they want you to answer it, I infer that they believe the df can be figured out. From the problem description, I can guess how many variables are being used to predict amount of government spending & the number of of total data. May 11, 2013 at 15:59
• k = 2, since there is tax revenue and level of nationwide debt. But how do we find out what n is? May 11, 2013 at 16:06
• How many data points are there? I suspect the answer is in the problem statement. May 11, 2013 at 16:10
• Omg is it 2008-1958 = 50? Oh dear..is that it? May 11, 2013 at 16:13

The question wants to find out R-sq

Rsq = Variance of Variables in the model/Total variance

Variance = MSE/df; hence df is required.

let n be number of years of observation =51. Let k be number of variables (treatments) in the model = 2 (tax revenue, natl debt)

Hence; SSTr = MSTr/treatment df = (12200-15)/(51-2)

And

Rsq = {(12200-15)/(51-2)}/(12200/51-1)