1. **State the problem:** We need to find the correlation coefficient $r$ for the given data points $(x, y)$: $(0, 15), (5, 10), (10, 5), (15, 0)$. 2. **Formula for correlation coefficient:** $$r = \frac{n\sum xy - \sum x \sum y}{\sqrt{\left(n\sum x^2 - (\sum x)^2\right)\left(n\sum y^2 - (\sum y)^2\right)}}$$ where $n$ is the number of data points. 3. **Calculate sums:** - $n = 4$ - $\sum x = 0 + 5 + 10 + 15 = 30$ - $\sum y = 15 + 10 + 5 + 0 = 30$ - $\sum xy = (0)(15) + (5)(10) + (10)(5) + (15)(0) = 0 + 50 + 50 + 0 = 100$ - $\sum x^2 = 0^2 + 5^2 + 10^2 + 15^2 = 0 + 25 + 100 + 225 = 350$ - $\sum y^2 = 15^2 + 10^2 + 5^2 + 0^2 = 225 + 100 + 25 + 0 = 350$ 4. **Substitute into formula:** $$r = \frac{4(100) - (30)(30)}{\sqrt{\left(4(350) - 30^2\right)\left(4(350) - 30^2\right)}}$$ 5. **Simplify numerator:** $$4(100) - 30 \times 30 = 400 - 900 = -500$$ 6. **Simplify denominator:** $$\sqrt{(1400 - 900)(1400 - 900)} = \sqrt{500 \times 500} = 500$$ 7. **Calculate $r$:** $$r = \frac{-500}{500} = -1$$ 8. **Interpretation:** The correlation coefficient is $-1$, indicating a perfect negative linear relationship between $x$ and $y$. **Final answer:** $$\boxed{-1}$$