Question at position 31 Which statements below about least-squares regression are correct? I. Switching the explanatory and response variables will not change the least-squares regression line. II. The slope of the line is very sensitive to outliers with large residuals. III. A value of r2 close to 1 does not guarantee that the relationship between the variables is linear.Only III is correctBoth II and III are correctI, II, and III are correctOnly I is correctOnly II is correct

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BlackTom AI · Accepted Answer

To begin, let's lay out the statements and options clearly so each can be evaluated on its own merits.
Option A: Only III is correct
Option B: Both II and III are correct
Option C: I, II, and III are correct
Option D: Only I is correct
Option E: Only II is correct

Now, assess each statement in turn, without jumping to a conclusion:
- I. Switching the explanatory and response variables will not change the least-squares regression line.
  This is not true in general. The least-squares line for y on x is not the same as the line for x on y. When you swap the roles of the variables, you’re fitting a different model with a different objective (minimizing the vertical residuals for y on x versus horizontal residuals for x on y), which yields a different slope and intercept. Therefore, I is false.
- II. The slope of the line is very sensitive to outliers with large residuals.
  Outliers that have large residuals can exert substantial leverage, pulling the regression line toward themselves and altering the slope notably. This sensitivity is a well-known property of ordinary least squares regression, so II is true.
- III. A value of r2 close to 1 does not guarantee that the relationship between the variables is linear.
  R-squared measures how well the linear model explains the variability in the data, given the linear form. A high r2 can occur even in some nonlinear patterns if the data align closely with a straight line in the observed range, or due to peculiar sample characteristics. Thus, r2 near 1 does not by itself certify linearity of the true relationship, making III true as a caveat.

With these evaluations in mind, you can compare the options:
- If only III were correct, that would imply I and II are false, which contradicts the established truth about II; so this option is incorrect.
- If both II and III are correct, that aligns with the assessments above, making this option plausible.
- Claiming I, II, and III are correct would duplicate I’s falsehood, so this option is incorrect.
- Saying Only I is correct is inconsistent with II being true and I being false, so this option is incorrect.
- Saying Only II is correct ignores the validity of III, which we've shown to be true, so this option is incorrect.

In summary, the reasoning shows that I is false, II is true, and III is true, which aligns with the option that states Both II and III are correct. No need to rely on any external tricks—the conclusions follow directly from the definitions and properties of least-squares regression, as discussed above.

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