To determine the slope of the least squares regression line, we need to look at the correlation coefficient [tex]\( r \)[/tex]. The correlation coefficient [tex]\( r \)[/tex] indicates the direction and strength of a linear relationship between two variables.
### Part (a)
Given: [tex]\( r = 0.7 \)[/tex]
- The correlation coefficient [tex]\( r \)[/tex] is positive.
- Since [tex]\( r \)[/tex] is positive, the slope of the regression line is also positive.
Answer: The slope of the line is positive.
Correct Answer: C. The slope of the line is positive
### Part (b)
Given: [tex]\( r = -0.7 \)[/tex]
- The correlation coefficient [tex]\( r \)[/tex] is negative.
- Since [tex]\( r \)[/tex] is negative, the slope of the regression line is negative.
Answer: The slope of the line is negative.
Correct Answer: B. The slope of the line is negative
### Part (c)
Given: [tex]\( r = 0 \)[/tex]
- The correlation coefficient [tex]\( r \)[/tex] is zero.
- Since [tex]\( r \)[/tex] is zero, there is no linear relationship between the variables, so the slope of the regression line is zero.
Answer: The slope of the line is zero.
Correct Answer: A. The slope of the line is 0
### Part (d)
Given: [tex]\( r^2 = 0.36 \)[/tex]
- The value [tex]\( r^2 \)[/tex] is the coefficient of determination.
- To find [tex]\( r \)[/tex], we take the square root of [tex]\( r^2 \)[/tex]:
[tex]\[
r = \pm \sqrt{0.36} = \pm 0.6
\][/tex]
- Here, [tex]\( r \)[/tex] could be either positive 0.6 or negative 0.6.
- Therefore, the slope of the regression line can be either positive or negative.
Answer: The slope of the line can be positive or negative.
Correct Answer: D. The slope of the line can be positive or negative
