To calculate the value of \(a\) (the Sum of Squares for Treatments), we need to use the total Sum of Squares (SS) and the Sum of Squares for Blocks and Error. The relationship between these sums of squares is given by:
\[ \text{Total SS} = \text{Treatment SS} + \text{Block SS} + \text{Error SS} \]
Given values:
- Total SS = 935.75
- Block SS = 375.25 (which is given in the table)
- Error SS = \(b\) (which we will find out later)
Substituting the known values into the equation, we can express it as:
\[ 935.75 = a + 375.25 + b \]
Now, let's find \(b\). The degrees of freedom (df) for Error is 9, and we know that:
\[ \text{Error MS} = \frac{\text{Error SS}}{\text{Error df}}. \]
The Mean Square Error (MS Error) is given in the table as 8.14. Thus, we have:
\[ 8.14 = \frac{b}{9}. \]
Now, solving for \(b\):
\[ b = 8.14 \times 9 = 73.26. \]
Now that we have \(b\), we can substitute it back into the first equation:
\[ 935.75 = a + 375.25 + 73.26. \]
Calculating the right side:
\[ 375.25 + 73.26 = 448.51. \]
Now substituting that back into the equation:
\[ 935.75 = a + 448.51. \]
Solving for \(a\):
\[ a = 935.75 - 448.51 = 487.24. \]
Thus, the value of \(a\) (the Sum of Squares for Treatments) is:
\[ \boxed{487.24}. \]
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