Question

what is the relationship between the linear correlation coefficient r and the slope b₁ of a regression line?

choose the correct answer below.
a. the value of r will always be larger than the value of b₁.
b. the value of r will always have the opposite sign of the value of b₁.
c. the value of r will always have the same sign as the value of b₁.
d. the value of r will always be smaller than the value of b₁.

Explanation:

To determine the relationship between the linear correlation coefficient \( r \) and the slope \( b_1 \) of a regression line, we analyze the sign (positive or negative) of each:

• The linear correlation coefficient \( r \) measures the strength and direction of a linear relationship. A positive \( r \) indicates a positive linear association, and a negative \( r \) indicates a negative linear association.
• The slope \( b_1 \) of the regression line represents the rate of change in the dependent variable for a unit change in the independent variable. A positive \( b_1 \) means the line rises (positive association), and a negative \( b_1 \) means the line falls (negative association).

Thus, the sign of \( r \) (direction of correlation) and the sign of \( b_1 \) (direction of the regression line’s slope) will always match. For example:

• If \( r > 0 \) (positive correlation), \( b_1 > 0 \) (positive slope).
• If \( r < 0 \) (negative correlation), \( b_1 < 0 \) (negative slope).

Options A and D are incorrect because \( r \) and \( b_1 \) are not constrained by magnitude (e.g., \( r \) ranges between \(-1\) and \(1\), but \( b_1 \) can be any real number). Option B is incorrect because their signs are the same, not opposite.

Answer:

C. The value of \( r \) will always have the same sign as the value of \( b_1 \).