1. **Stating the problem:** We have two sets of data points: $X_i$ and $Y_i$ with 15 values each. The goal is to analyze the relationship between $X_i$ and $Y_i$. 2. **Formula used:** To understand the relationship, we can calculate the Pearson correlation coefficient $r$ which measures linear correlation between two variables: $$r = \frac{n\sum X_iY_i - \sum X_i \sum Y_i}{\sqrt{\left(n\sum X_i^2 - (\sum X_i)^2\right)\left(n\sum Y_i^2 - (\sum Y_i)^2\right)}}$$ where $n=15$ is the number of data points. 3. **Calculate sums:** - $\sum X_i = 20+41+18+40+43+15+17+44+41+27+38+35+15+37+22 = 488$ - $\sum Y_i = 14+31+20+30+32+18+16+32+31+24+29+25+12+30+21 = 365$ - $\sum X_i^2 = 20^2+41^2+18^2+40^2+43^2+15^2+17^2+44^2+41^2+27^2+38^2+35^2+15^2+37^2+22^2 = 17454$ - $\sum Y_i^2 = 14^2+31^2+20^2+30^2+32^2+18^2+16^2+32^2+31^2+24^2+29^2+25^2+12^2+30^2+21^2 = 9313$ - $\sum X_iY_i = 20\times14 + 41\times31 + 18\times20 + 40\times30 + 43\times32 + 15\times18 + 17\times16 + 44\times32 + 41\times31 + 27\times24 + 38\times29 + 35\times25 + 15\times12 + 37\times30 + 22\times21 = 12344$ 4. **Plug values into formula:** $$r = \frac{15\times12344 - 488\times365}{\sqrt{(15\times17454 - 488^2)(15\times9313 - 365^2)}}$$ Calculate numerator: $$15\times12344 = 185160$$ $$488\times365 = 178120$$ $$\text{Numerator} = 185160 - 178120 = 7040$$ Calculate denominator parts: $$15\times17454 = 261810$$ $$488^2 = 238144$$ $$15\times9313 = 139695$$ $$365^2 = 133225$$ Calculate denominator: $$\sqrt{(261810 - 238144)(139695 - 133225)} = \sqrt{23666 \times 6470} = \sqrt{153095020} \approx 12374.6$$ 5. **Calculate correlation coefficient:** $$r = \frac{7040}{12374.6} \approx 0.569$$ 6. **Interpretation:** The correlation coefficient $r \approx 0.569$ indicates a moderate positive linear relationship between $X_i$ and $Y_i$. **Final answer:** The Pearson correlation coefficient between $X_i$ and $Y_i$ is approximately $0.569$.