Calculate R^2 of the regression line for predicting height from shoulder girth, and interpret it in the context of the application.

Question

BlackTom AI · Accepted Answer

First, let's lay out the problem: we are asked to calculate R^2 for predicting height from shoulder girth and then interpret what that R^2 means in the context of the application.
Option 1 states R^2 = -0.44 and that approximately 44% of the variation is explained by the model. A negative R^2 is not possible for a standard linear regression fit to data; R^2 ranges from 0 to 1 (though in some nonstandard cases it can be negative, it usually indicates a poor fit only if explicitly allowed, but here the phrasing implies a normal regression context). This makes the option inconsistent with the fundamental definition of R^2 as the proportion of explained variance, so it is incorrect.
Option 2 claims R^2 = 0.44 but then says approximately 56% of the variation is explained by the model. By definition, R^2 is the fraction of total variance explained by the model, so if R^2 is 0.44, about 44% is explained, and 56% remains unexplained. This option mixes the correct R^2 value with an incorrect interpretation of the remainder, so it is false.
Option 3 gives R^2 = 0.67 with an interpretation that 67% of the variation is explained by the model. Although the interpretation matches the numeric value, the numeric value itself is incorrect relative to the given correct answer. If the actual computed R^2 is 0.44, this option misstates both the R^2 value and the explained variance, making it incorrect.
Option 4 repeats R^2 = 0.44 and correctly states that approximately 44% of the variation in heights is accounted for by the model, i.e., explained by shoulder girth. This aligns with the standard interpretation: R^2 tells us the proportion of variance in the dependent variable (height) explained by the predictor (shoulder girth). Therefore, this option provides both the correct numeric value and the correct interpretation.
In summary, the key distinctions are: a correct R^2 cannot be negative for this context, the explained variance must match the R^2 value, and the option should pair the right numeric R^2 with the corresponding interpretation. The only option that satisfies all of these is Option 4.

Calculate R^2 of the regression line for predicting height from shoulder girth, and interpret it in the context of the application.单项选择题

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