Suppose that a simple regression has Yi is the dependent variable, X1i is the independent variable and subscript i runs over observations i=1,2,...,n. is the error term. The value of coefficient of determination is calculated as R2=0.01. Choose ALL CORRECT statement(s) about this low value of R2:

Question

BlackTom AI · Accepted Answer

R^2 is the fraction of the total variation in Y explained by the regression. In a simple regression the key identities are:
- TSS (total sum of squares) = ESS (explained/regression sum of squares) + SSE (sum of squared residuals, i.e. unexplained).
- R^2 = ESS / TSS = 1 − (SSE / TSS).

Step-by-step reasoning and intermediate calculations:
1. Given R^2 = 0.01, ESS / TSS = 0.01. That means only 1% of Y’s variance is explained by X1.
2. Using R^2 = 1 − (SSE / TSS), we get SSE / TSS = 1 − 0.01 = 0.99. So 99% of the total variance remains in the residuals (unexplained by X1).
3. Numerical illustration: if TSS = 100, then ESS = 0.01 × 100 = 1 and SSE = 0.99 × 100 = 99. This shows the regression explains very little of Y’s variation.

Why the correct answer is correct:
- Option b ("it suggests that there are other important factors to influence the left-hand side variable") is correct because such a low R^2 indicates X1 explains almost none of Y’s variance; therefore other variables or factors must account for most of the variation.

Why the other options are incorrect:
- a is incorrect: R^2 = 0.01 indicates poor—not good—explanatory power.
- c is incorrect given the option’s wording that SSR is the sum of squared residuals: a low R^2 implies the sum of squared residuals (SSE) is large relative to TSS (SSE/TSS = 0.99), not low. (If SSR were used in the alternative convention to mean regression sum of squares, then regression SSR would be low; but the option explicitly labels SSR as residuals.)
- d is incorrect: R^2 quantifies how much variance is explained, but it does not identify which other factors cause the remaining variation.

Conclusion: only option b is correct, which matches the provided answer.

类似问题

Which of the following is the correct interpretation for 0.60 coefficient of determination?