In a study comparing four groups with six observations in each group,the MSE = 487 and the MSG = 3456.What is the value of the coefficient of determination?

A) 0.484
B) 0.516
C) 0.587
D) 0.939

Question

In a study comparing four groups with six observations in each group,the MSE = 487 and the MSG = 3456.What is the value of the coefficient of determination?A) 0.484B) 0.516C) 0.587D) 0.939

Candice Rimas · Accepted Answer

Final Answer : B, Explanation :   The coefficient of determination,   R 2 R^2 R 2  , is calculated as   R 2 = S S G S S T R^2 = \frac{SSG}{SST} R 2 = SST SSG ​  , where   S S G SSG SSG   is the sum of squares between groups and   S S T SST SST   is the total sum of squares.   S S T = S S G + S S E SST = SSG + SSE SST = SSG + SSE  , where   S S E SSE SSE   is the sum of squares within groups (error). Given   M S E = 487 MSE = 487 MSE = 487   and   M S G = 3456 MSG = 3456 MSG = 3456  , with 4 groups of 6 observations each, we have   n = 24 n = 24 n = 24   total observations and   k = 4 k = 4 k = 4   groups.   S S E = M S E × ( n − k ) = 487 × ( 24 − 4 ) = 487 × 20 SSE = MSE 	imes (n - k) = 487 	imes (24 - 4) = 487 	imes 20 SSE = MSE × ( n − k ) = 487 × ( 24 − 4 ) = 487 × 20  .   S S G = M S G × ( k − 1 ) = 3456 × 3 SSG = MSG 	imes (k - 1) = 3456 	imes 3 SSG = MSG × ( k − 1 ) = 3456 × 3  . Thus,   R 2 = S S G S S G + S S E = 3456 × 3 3456 × 3 + 487 × 20 R^2 = \frac{SSG}{SSG + SSE} = \frac{3456 	imes 3}{3456 	imes 3 + 487 	imes 20} R 2 = SSG + SSE SSG ​ = 3456 × 3 + 487 × 20 3456 × 3 ​  . Calculating this gives   R 2 ≈ 0.516 R^2 \approx 0.516 R 2 ≈ 0.516  , indicating that approximately 51.6% of the variability is explained by the group differences.

In a study comparing four groups with six observations in each group,the MSE = 487 and the MSG = 3456.What is the value of the coefficient of determination?

A) 0.484
B) 0.516
C) 0.587
D) 0.939

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