Q.4.4. Linear Correlation - Elementary Statistics

Question

The coefficient of determination of a set of data points is $0.716$ .

a. Can you determine the linear correlation coefficient? If yes, obtain it. If no, why not?

b. Can you determine whether the slope of the regression line is positive or negative? Why or why not?

c. If we tell you that the slope of the regression line is negative, can you determine the linear correlation coefficient? If yes, obtain it. If no, why not?

d. If we tell you that the slope of the regression line is positive, can you determine the linear correlation coefficient? If yes, obtain it. If no, why not?

Step-by-Step Solution

Answer

(a) We are not supposed to determine the linear correlation coefficient $r$ even though we have the value of the coefficient of determination $r^2$

(b) We are not required to identify the sign of the slope of the regression line even though we have the value of the coefficient of determination $\left(r^2\right)$

(c)  It is impossible to calculate the linear correlation coefficient. 

(d)  It is impossible to calculate the linear correlation coefficient. 

Part (a)Step 1: Given information

Given in the question that, The coefficient of determination of a set of data points is$0.716$ . We need to determine the linear correlation coefficient

Part(a) Step 2: Explanation

A set of data points has a coefficient of determination of $0.716$.

The coefficient of determination for a set of data points is$r^2=0.716$in this case.

We are not supposed to determine the linear correlation coefficient $\left(r\right)$ even though we have the value of the coefficient of determination $r^2$ because the coefficient of determination cannot tell us whether there is a linear relationship between the supplied variables.

Part(b) Step 1: Given information

Given in the question that, The coefficient of determination of a set of data points is $0.716$ .  We need to determine that whether the slope of the regression line is positive or negative

Part (b) Step 2: Explanation

A set of data points has a coefficient of determination of $0.716$.

We are not required to identify the sign of the slope of the regression line even though we have the value of the coefficient of determination $\left(r^2\right)$ because $r=\pm \sqrt{r^2}$.

Part(c) Step 1: Given information

Given in the question that, The coefficient of determination of a set of data points is $0.716$ .  We need to determine the linear correlation coefficient if  the slope of the regression line is negative.

Part(c) Step 2: Explanation

A set of data points has a coefficient of determination of $0.716$.

We don't know if there is a linear relationship between the variables because the coefficient of determination is negative and the slope sign is negative. As a result, it is impossible to calculate the linear correlation coefficient.

Part(d) Step 1: Given information

Given in the question that, The coefficient of determination of a set of data points is $0.716$ . We need to determine the linear correlation coefficient  if  the slope of the regression line is positive.

Part(d) Step 2: Explanation

A set of data points has a coefficient of determination of $0.716$.

Even though the coefficient of determination is positive and the slope is positive, we are unsure if there is a linear relationship between the variables. As a result, it is impossible to calculate the linear correlation coefficient.