1. **Problem Statement:** Given two sets of data points for variables $x$ and $y$: $$x = \{6,8,10,12,14,17,18,20\}$$ $$y = \{15,20,17,21,27,25,32,37\}$$ Calculate the correlation coefficient between $x$ and $y$ to understand the strength and direction of their linear relationship. 2. **Formula Used:** The Pearson correlation coefficient $r$ is given by: $$r = \frac{n\sum xy - \sum x \sum y}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}}$$ where $n$ is the number of data points. 3. **Calculate sums and sums of squares:** - $n = 8$ - $\sum x = 6 + 8 + 10 + 12 + 14 + 17 + 18 + 20 = 105$ - $\sum y = 15 + 20 + 17 + 21 + 27 + 25 + 32 + 37 = 194$ - $\sum x^2 = 6^2 + 8^2 + 10^2 + 12^2 + 14^2 + 17^2 + 18^2 + 20^2 = 36 + 64 + 100 + 144 + 196 + 289 + 324 + 400 = 1553$ - $\sum y^2 = 15^2 + 20^2 + 17^2 + 21^2 + 27^2 + 25^2 + 32^2 + 37^2 = 225 + 400 + 289 + 441 + 729 + 625 + 1024 + 1369 = 5102$ - $\sum xy = (6)(15) + (8)(20) + (10)(17) + (12)(21) + (14)(27) + (17)(25) + (18)(32) + (20)(37) = 90 + 160 + 170 + 252 + 378 + 425 + 576 + 740 = 2791$ 4. **Substitute values into the formula:** $$r = \frac{8 \times 2791 - 105 \times 194}{\sqrt{(8 \times 1553 - 105^2)(8 \times 5102 - 194^2)}}$$ Calculate numerator: $$8 \times 2791 = 22328$$ $$105 \times 194 = 20370$$ $$\text{Numerator} = 22328 - 20370 = 1958$$ Calculate denominator parts: $$8 \times 1553 = 12424$$ $$105^2 = 11025$$ $$8 \times 5102 = 40816$$ $$194^2 = 37636$$ Calculate denominator: $$\sqrt{(12424 - 11025)(40816 - 37636)} = \sqrt{1399 \times 3180} = \sqrt{4445820} \approx 2108.52$$ 5. **Calculate correlation coefficient:** $$r = \frac{1958}{2108.52} \approx 0.9289$$ 6. **Interpretation:** The correlation coefficient $r \approx 0.93$ indicates a strong positive linear relationship between $x$ and $y$. **Final answer:** $$\boxed{r \approx 0.93}$$