To address this problem, we will determine two key statistical measures: the correlation coefficient [tex]\( r \)[/tex] and the coefficient of determination [tex]\( R^2 \)[/tex].
1. Correlation Coefficient [tex]\( r \)[/tex]:
The correlation coefficient [tex]\( r \)[/tex] measures the strength and direction of the linear relationship between two variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. It ranges from -1 to 1, where:
- [tex]\( r = 1 \)[/tex] indicates a perfect positive linear relationship,
- [tex]\( r = -1 \)[/tex] indicates a perfect negative linear relationship,
- [tex]\( r = 0 \)[/tex] indicates no linear relationship.
Based on our calculations, the correlation coefficient [tex]\( r \)[/tex] for the given data set is:
[tex]\[
r = 0.519
\][/tex]
2. Coefficient of Determination [tex]\( R^2 \)[/tex]:
The coefficient of determination [tex]\( R^2 \)[/tex] indicates the proportion of the variance in the dependent variable [tex]\( y \)[/tex] that is predictable from the independent variable [tex]\( x \)[/tex]. It is calculated by squaring the correlation coefficient [tex]\( r \)[/tex]:
[tex]\[
R^2 = r^2
\][/tex]
To express [tex]\( R^2 \)[/tex] as a percentage, we multiply by 100:
[tex]\[
R^2 (\%) = r^2 \times 100
\][/tex]
Given [tex]\( r = 0.519 \)[/tex]:
[tex]\[
R^2 \approx (0.519)^2 \times 100 = 0.269 \times 100 = 26.9 \%
\][/tex]
In summary, the correlation coefficient [tex]\( r \)[/tex] is [tex]\( 0.519 \)[/tex], and the proportion of the variation in [tex]\( y \)[/tex] that can be explained by the variation in [tex]\( x \)[/tex] is [tex]\( 26.9\% \)[/tex].
To formally answer the questions:
1. The correlation coefficient accurate to three decimal places is:
[tex]\[
r = 0.519
\][/tex]
2. The proportion of the variation in [tex]\( y \)[/tex] explained by the variation in [tex]\( x \)[/tex] accurate to one decimal place is:
[tex]\[
R^2 = 26.9\%
\][/tex]
