1. **Problem Statement:** Determine if there is a significant relationship between the scores of students in Physics and Math subjects using the given data. 2. **Data:** Physics scores: $14, 10, 8, 12, 10, 8, 7, 6, 5, 0$ Math scores: $10, 10, 8, 6, 3, 12, 13, 14, 16, 18$ 3. **Method:** Use Pearson's correlation coefficient $r$ to measure the strength and direction of the linear relationship between the two subjects. 4. **Formula for Pearson's correlation coefficient:** $$r = \frac{n\sum xy - \sum x \sum y}{\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)}}$$ where $x$ and $y$ are the scores in Physics and Math respectively, and $n$ is the number of pairs. 5. **Calculate sums:** - $n = 10$ - $\sum x = 14 + 10 + 8 + 12 + 10 + 8 + 7 + 6 + 5 + 0 = 80$ - $\sum y = 10 + 10 + 8 + 6 + 3 + 12 + 13 + 14 + 16 + 18 = 110$ - $\sum xy = (14)(10) + (10)(10) + (8)(8) + (12)(6) + (10)(3) + (8)(12) + (7)(13) + (6)(14) + (5)(16) + (0)(18) = 140 + 100 + 64 + 72 + 30 + 96 + 91 + 84 + 80 + 0 = 757$ - $\sum x^2 = 14^2 + 10^2 + 8^2 + 12^2 + 10^2 + 8^2 + 7^2 + 6^2 + 5^2 + 0^2 = 196 + 100 + 64 + 144 + 100 + 64 + 49 + 36 + 25 + 0 = 778$ - $\sum y^2 = 10^2 + 10^2 + 8^2 + 6^2 + 3^2 + 12^2 + 13^2 + 14^2 + 16^2 + 18^2 = 100 + 100 + 64 + 36 + 9 + 144 + 169 + 196 + 256 + 324 = 1398$ 6. **Calculate numerator:** $$n\sum xy - \sum x \sum y = 10 \times 757 - 80 \times 110 = 7570 - 8800 = -1230$$ 7. **Calculate denominator:** $$\sqrt{(n\sum x^2 - (\sum x)^2)(n\sum y^2 - (\sum y)^2)} = \sqrt{(10 \times 778 - 80^2)(10 \times 1398 - 110^2)}$$ $$= \sqrt{(7780 - 6400)(13980 - 12100)} = \sqrt{1380 \times 1880} = \sqrt{2594400} \approx 1610.11$$ 8. **Calculate $r$:** $$r = \frac{-1230}{1610.11} \approx -0.7637$$ 9. **Degrees of freedom:** $df = n - 2 = 10 - 2 = 8$ 10. **Tabular critical value at 0.05 significance level for $df=8$:** $0.632$ 11. **Decision:** Since $|r| = 0.7637 > 0.632$, the correlation is significant at the 0.05 level. 12. **Conclusion:** There is a significant negative relationship between the Physics and Math scores.