Question at position 11 The coefficient of determination ………….. (Select all that apply)is used to evaluate the goodness of fitis equal to zero for a perfect fittakes values between –1 to +1is equal to negative one for the poorest fit

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BlackTom AI · Accepted Answer

The question asks about the coefficient of determination and its properties. 
Option 1: 'is used to evaluate the goodness of fit' — This is accurate. The coefficient of determination, commonly denoted R^2, quantifies how well the model explains the variability of the response data; higher values indicate a better fit and it is a standard measure of goodness of fit. 
Option 2: 'is equal to zero for a perfect fit' — This is incorrect. A perfect fit yields R^2 = 1, meaning all variability is explained by the model; R^2 = 0 would indicate no explanatory power at all. 
Option 3: 'takes values between –1 to +1' — This statement is false for R^2. R^2 is defined to be between 0 and 1 in ordinary least squares contexts. The confusion often comes from the correlation coefficient, r, which does lie between –1 and +1, but R^2 is the square of that value and nonnegative. 
Option 4: 'is equal to negative one for the poorest fit' — This is incorrect. The poorest fit does not give R^2 = –1; in fact, R^2 cannot be negative in standard formulations (it ranges from 0 to 1). In some extended or adjusted contexts, related metrics can be negative, but plain R^2 is nonnegative, with zero representing no explanatory power and one representing a perfect fit.

Question at position 11 The coefficient of determination ………….. (Select all that apply)is used to evaluate the goodness of fitis equal to zero for a perfect fittakes values between –1 to +1is equal to negative one for the poorest fit

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