To find the correlation coefficient for the data provided in the table, we need to measure the strength and direction of the linear relationship between the two variables, [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
The data points are as follows:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
0 & 15 \\
\hline
5 & 10 \\
\hline
10 & 5 \\
\hline
15 & 0 \\
\hline
\end{tabular}
\][/tex]
The correlation coefficient [tex]\(\rho\)[/tex] (or [tex]\(r\)[/tex]) ranges from -1 to 1. It measures the strength and direction of a linear relationship:
- [tex]\(\rho = 1\)[/tex] indicates a perfect positive linear relationship.
- [tex]\(\rho = -1\)[/tex] indicates a perfect negative linear relationship.
- [tex]\(\rho = 0\)[/tex] indicates no linear relationship.
To determine the correlation coefficient, we follow the formula for Pearson's correlation coefficient, but I've already determined that the numerical result is [tex]\(-1.0\)[/tex]. This result indicates the following:
- The data exhibit a perfect negative linear relationship.
- As [tex]\(x\)[/tex] increases, [tex]\(y\)[/tex] decreases in a perfectly linear fashion.
Thus, the correlation coefficient for the given data is:
[tex]\[
\boxed{-1}
\][/tex]
