1. **State the problem:** We are given scores of 20 students in Mathematics and Physics and asked to find the relationship using the Pearson product-moment correlation coefficient. Then, we need to interpret the result. 2. **Recall Pearson correlation formula:** $$r = \frac{n\sum xy - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}$$ where $x$ is Mathematics scores, $y$ is Physics scores, and $n=20$. 3. **List the scores:** - Mathematics $x$: 98, 97, 95, 94, 93, 91, 90, 89, 88, 87, 86, 84, 83, 81, 80, 79, 77, 76, 75, 74 - Physics $y$: 76, 75, 72, 70, 68, 66, 44, 60, 58, 57, 56, 54, 52, 50, 48, 46, 44, 45, 62, 77 4. **Calculate sums:** $$\sum x = 1662$$ $$\sum y = 1083$$ $$\sum x^2 = 139470$$ $$\sum y^2 = 61547$$ $$\sum xy = 88245$$ 5. **Substitute values into formula:** $$r = \frac{20 \times 88245 - (1662)(1083)}{\sqrt{[20 \times 139470 - 1662^2][20 \times 61547 - 1083^2]}}$$ 6. **Calculate numerator:** $$20 \times 88245 = 1764900$$ $$(1662)(1083) = 1798746$$ $$\text{Numerator} = 1764900 - 1798746 = -33846$$ 7. **Calculate denominator:** $$20 \times 139470 = 2789400$$ $$1662^2 = 2763044$$ $$20 \times 61547 = 1230940$$ $$1083^2 = 1172889$$ Calculate inside square roots: $$2789400 - 2763044 = 26356$$ $$1230940 - 1172889 = 58051$$ Therefore, $$\sqrt{26356 \times 58051} = \sqrt{1529889256} \approx 39108.67$$ 8. **Calculate $r$:** $$r = \frac{-33846}{39108.67} \approx -0.865\n$$ 9. **Interpretation:** The correlation coefficient $r \approx -0.865$ indicates a strong negative linear relationship between Mathematics and Physics scores. This means as Mathematics scores increase, Physics scores tend to decrease, and vice versa. 10. **Judgement:** There is a strong inverse relationship between the performances in Mathematics and Physics in this data set, which is unusual and may warrant further investigation regarding the data or context.