Answer :
To calculate the correlation coefficient, we will use the formula for Pearson's correlation coefficient, denoted as [tex]r[/tex]. Pearson's correlation coefficient measures the strength and direction of a linear relationship between two variables on a scatterplot.
Given:
[tex]x: \{5, 6, 7, 8, 9, 10\}[/tex]
[tex]y: \{44, 39, 39, 36, 31, 28\}[/tex]
Step-by-step Calculation:
Calculate the means of [tex]x[/tex] and [tex]y[/tex]:
(
\bar{x} = \frac{5 + 6 + 7 + 8 + 9 + 10}{6} = 7.5
)(
\bar{y} = \frac{44 + 39 + 39 + 36 + 31 + 28}{6} = 36.1667
)Calculate the components for the formula:
- Sum of the product of the deviations of [tex]x[/tex] and [tex]y[/tex]:
[tex]\sum{(x_i - \bar{x})(y_i - \bar{y})}[/tex]
- Sum of the squares of the deviations of [tex]x[/tex]:
[tex]\sum{(x_i - \bar{x})^2}[/tex]
- Sum of the squares of the deviations of [tex]y[/tex]:
[tex]\sum{(y_i - \bar{y})^2}[/tex]
Calculate the correlation coefficient [tex]r[/tex]:
[tex]r = \frac{\sum{(x_i - \bar{x})(y_i - \bar{y})}}{\sqrt{\sum{(x_i - \bar{x})^2} \cdot \sum{(y_i - \bar{y})^2}}}[/tex]
Final Answer:
After calculating the sum of products and deviations:
- [tex]\sum{(x_i - \bar{x})(y_i - \bar{y})} = -49.5[/tex]
- [tex]\sum{(x_i - \bar{x})^2} = 17.5[/tex]
- [tex]\sum{(y_i - \bar{y})^2} = 187.67[/tex]
Plugging these into the formula gives:
[tex]r = \frac{-49.5}{\sqrt{17.5 \cdot 187.67}} \approx -0.901[/tex]
Thus, the correlation coefficient [tex]r[/tex] is approximately [tex]-0.901[/tex], indicating a strong negative linear relationship between the variables.