To determine which table shows a negative correlation between the [tex]\( x \)[/tex] values and the [tex]\( y \)[/tex] values, we examine the correlation coefficients for each table:
1. First table:
- [tex]\( x \)[/tex]: [tex]\([2, 5, 6, 7, 10, 12]\)[/tex]
- [tex]\( y \)[/tex]: [tex]\([-8, -5, -6, -3, -2, -1]\)[/tex]
- Correlation coefficient: [tex]\(0.9530704598482771\)[/tex]
2. Second table:
- [tex]\( x \)[/tex]: [tex]\([2, 5, 6, 7, 10, 12]\)[/tex]
- [tex]\( y \)[/tex]: [tex]\([-5, -5, -5, -5, -5, -5]\)[/tex]
- Correlation coefficient: [tex]\(nan\)[/tex]
- Note: This result occurs because the [tex]\( y \)[/tex] values are constant, so there is no meaningful correlation.
3. Third table:
- [tex]\( x \)[/tex]: [tex]\([2, 5, 6, 7, 10, 12]\)[/tex]
- [tex]\( y \)[/tex]: [tex]\([6, 3, 1, 1, 3, 6]\)[/tex]
- Correlation coefficient: [tex]\(0.049669963389939197\)[/tex]
4. Fourth table:
- [tex]\( x \)[/tex]: [tex]\([2, 5, 6, 7, 10, 12]\)[/tex]
- [tex]\( y \)[/tex]: [tex]\([4, 2, -4, -3, -11, -12]\)[/tex]
- Correlation coefficient: [tex]\(-0.9655651219223368\)[/tex]
The correlation coefficient ranges from -1 to 1:
- A value of 1 indicates a perfect positive correlation.
- A value of -1 indicates a perfect negative correlation.
- Values close to 0 indicate no linear correlation.
Among the tables provided, the fourth table has a correlation coefficient of [tex]\( -0.9655651219223368 \)[/tex], which indicates a strong negative correlation.
Therefore, the fourth table shows a negative correlation.
