To determine the type of correlation between the variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex] given in the data set, we need to compute the correlation coefficient. This coefficient quantifies the degree to which two variables are related.
The data set provided is:
[tex]\[
\begin{tabular}{l|cccc}
$x$ & 1 & 2 & 3 & 4 \\
\hline
$y$ & 10 & 8 & 4 & -2
\end{tabular}
\][/tex]
The correlation coefficient, often denoted as [tex]\( r \)[/tex], can vary between -1 and 1:
- If [tex]\( r \)[/tex] is close to 1, there is a strong positive correlation.
- If [tex]\( r \)[/tex] is close to -1, there is a strong negative correlation.
- If [tex]\( r \)[/tex] is close to 0, there is no correlation.
For our data set, the correlation coefficient [tex]\( r \)[/tex] is calculated to be approximately -0.9759.
To interpret this:
1. Magnitude: The coefficient is close to 1 in magnitude, which indicates a strong correlation.
2. Sign: The coefficient is negative (-0.9759), indicating that as [tex]\( x \)[/tex] increases, [tex]\( y \)[/tex] tends to decrease.
Given this strong negative coefficient, we can conclude that the type of correlation observed in the data set is:
A. Negative
