Answer :
To compute the correlation coefficient between the two sets of data, [tex]x[/tex] and [tex]y[/tex], we will use Pearson's correlation formula, which is a measure of the linear relationship between two sets of data.
The formula for Pearson's correlation coefficient [tex]r[/tex] is given by:
[tex]r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n \sum x^2 - (\sum x)^2][n \sum y^2 - (\sum y)^2]}}[/tex]
Given:
- [tex]x: 77, 66, 66, 77, 99[/tex]
- [tex]y: 33, 77, 66, 99, 5[/tex]
Let's break down the steps:
Calculate Each Sum Needed:
[tex]\sum x = 77 + 66 + 66 + 77 + 99 = 385[/tex]
[tex]\sum y = 33 + 77 + 66 + 99 + 5 = 280[/tex]
Calculate Each Square Sum Needed:
[tex]\sum x^2 = 77^2 + 66^2 + 66^2 + 77^2 + 99^2 = 6455[/tex]
[tex]\sum y^2 = 33^2 + 77^2 + 66^2 + 99^2 + 5^2 = 1880[/tex]
Calculate the Sum of Products:
[tex]\sum xy = (77 \times 33) + (66 \times 77) + (66 \times 66) + (77 \times 99) + (99 \times 5) = 7582[/tex]
Substitute Into the Formula:
Using [tex]n = 5[/tex] (since there are 5 data points):
[tex]r = \frac{5(7582) - (385)(280)}{\sqrt{[5 \times 6455 - 385^2][5 \times 1880 - 280^2]}}[/tex]
[tex]r = \frac{37910 - 107800}{\sqrt{[32275 - 148225][9400 - 78400]}}[/tex]
[tex]r = \frac{37910 - 107800}{\sqrt{161375 \times 80400}}[/tex]
[tex]r = \frac{-69890}{\sqrt{20974050000}}[/tex]
[tex]r \approx \frac{-69890}{144850.86} \approx -0.48[/tex]
The correlation coefficient [tex]r \approx -0.48[/tex] indicates a moderate negative linear relationship between [tex]x[/tex] and [tex]y[/tex]. This suggests that as the values of [tex]x[/tex] increase, the values of [tex]y[/tex] tend to decrease.
