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Describe the slope of the least squares regression line for the given information:

a. [tex]r=0.7[/tex]
b. [tex]r=-0.7[/tex]
c. [tex]r=0[/tex]
d. [tex]r^2=0.36[/tex]

Choose the correct answer below:

A. The slope of the line is 0.
B. The slope of the line is negative.
C. The slope of the line is positive.
D. The slope of the line can be positive or negative.


Sagot :

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