Properties of the Standard Error of the Slope

Question

Researchers are studying the relationship between enzyme concentration, $$x$$, and reaction time in seconds, $$y$$. A linear regression analysis is performed, yielding an estimated slope $$b$$ and its standard error $$s_b$$. If the population variance of the reaction time, $$y$$, were to increase while the variability of the enzyme concentration, $$x$$, remained constant, what would be the effect on $$s_b$$ and the corresponding t-distribution for the slope?

Accepted Answer

An increase in the reaction time variance would raise the residual standard error, thereby increasing $$s_b$$ and widening the confidence interval, even though the degrees of freedom stay the same.

Answer

The t-distribution would adjust for the increased variance, leaving the standard error and confidence interval unchanged.

Answer

The increased variance in reaction time would have no effect on $$s_b$$ since it depends solely on the spread of $$x$$.

Answer

A higher reaction time variance would unexpectedly decrease $$s_b$$ by normalizing the error term, leading to a narrower confidence interval.

Properties of the Standard Error of the Slope

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