First, let's introduce the concept of the correlation coefficient. The correlation coefficient, often denoted as [tex]\( r \)[/tex], measures the strength and direction of a linear relationship between two variables. Its value ranges from [tex]\(-1\)[/tex] to [tex]\(1\)[/tex], where:
- [tex]\(1\)[/tex] indicates a perfect positive linear relationship.
- [tex]\(-1\)[/tex] indicates a perfect negative linear relationship.
- [tex]\(0\)[/tex] indicates no linear relationship.
Given the table of data, we have:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
0 & 15 \\
\hline
5 & 10 \\
\hline
10 & 5 \\
\hline
15 & 0 \\
\hline
\end{array}
\][/tex]
To determine the correlation coefficient [tex]\( r \)[/tex], follow these steps:
1. Calculate the means of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
- The mean of [tex]\( x \)[/tex] ([tex]\( \bar{x} \)[/tex]) = [tex]\(\frac{0 + 5 + 10 + 15}{4} = 7.5\)[/tex]
- The mean of [tex]\( y \)[/tex] ([tex]\( \bar{y} \)[/tex]) = [tex]\(\frac{15 + 10 + 5 + 0}{4} = 7.5\)[/tex]
2. Calculate the covariance of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
- Covariance formula: [tex]\[ \text{Cov}(x, y) = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y}) \][/tex]
3. Calculate the standard deviations of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
- Standard deviation formula: [tex]\[ \sigma_x = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2} \][/tex]
- [tex]\[ \sigma_y = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (y_i - \bar{y})^2} \][/tex]
4. Compute the correlation coefficient.
- Correlation coefficient formula: [tex]\[ r = \frac{\text{Cov}(x, y)}{\sigma_x \sigma_y} \][/tex]
After performing these calculations with the given data, we find that the correlation coefficient is:
[tex]\[ r = -1.0 \][/tex]
This indicates a perfect negative linear relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. Hence, the correct answer for the correlation coefficient for the data shown in the table is:
[tex]\[ -1 \][/tex]
