Question

Which of the following is the correct value of the correlation and its corresponding interpretation?

a. The correlation is $0.947$, and $94.7\%$of the variability in a bear’s weight can be accounted for by the least-squares regression line using neck girth as the explanatory variable.

b. The correlation is $0.947$. The linear association between a bear’s neck girth and its weight is strong and positive.

c. The correlation is $0.973$, and $97.3\%$of the variability in a bear’s weight can be accounted for by the least-squares regression line using neck girth as the explanatory variable.

d. The correlation is $0.973$. The linear association between a bear’s neck girth and its weight is strong and positive.

e. The correlation cannot be calculated without the data.

Short answer

Option (d) is the correct option.

Step-by-step solution

Step 1. Given information

a. The correlation is $0.947$, and $94.7\%$of the variability in a bear’s weight can be accounted for by the least-squares regression line using neck girth as the explanatory variable.

b. The correlation is $0.947$. The linear association between a bear’s neck girth and its weight is strong and positive.

c. The correlation is $0.973$, and $97.3\%$of the variability in a bear’s weight can be accounted for by the least-squares regression line using neck girth as the explanatory variable.

d. The correlation is $0.973$. The linear association between a bear’s neck girth and its weight is strong and positive.

e. The correlation cannot be calculated without the data.

Step 2. Explanation

Park rangers are interested in estimating the weight of the bears that inhabit their state. The output of this data is given in the question. The coefficient of determination$r^2$is given in the computer output following “R-Sq” as:

$r^2=94.7\%=0.947$

The correlation coefficient is the positive or negative square root of the coefficient of determination. The pattern in the scatterplot slopes upward which implies that there is a positive relationship between the variables and thus the correlation coefficient needs to be positive. Thus, it is calculated as:

$$\begin{gathered} r=+\sqrt{r^2} \\ =+\sqrt{0.947} \\ =0.973 \end{gathered}$$

Since the correlation coefficient is close to one, the linear relationship between the variables is strong and thus the option (d) is the correct option.