1. **State the problem:** We want to calculate the Pearson correlation coefficient $r$ between hours spent studying $X = [10,8,6,12,14]$ and exam scores $Y = [85,80,75,90,92]$ to see if there is a relationship. 2. **Formula:** The Pearson correlation coefficient 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:** - $n = 5$ - $\sum X = 10 + 8 + 6 + 12 + 14 = 50$ - $\sum Y = 85 + 80 + 75 + 90 + 92 = 422$ - $\sum XY = (10)(85) + (8)(80) + (6)(75) + (12)(90) + (14)(92) = 850 + 640 + 450 + 1080 + 1288 = 4308$ - $\sum X^2 = 10^2 + 8^2 + 6^2 + 12^2 + 14^2 = 100 + 64 + 36 + 144 + 196 = 540$ - $\sum Y^2 = 85^2 + 80^2 + 75^2 + 90^2 + 92^2 = 7225 + 6400 + 5625 + 8100 + 8464 = 35814$ 4. **Plug into formula numerator:** $$n\sum XY - \sum X \sum Y = 5 \times 4308 - 50 \times 422 = 21540 - 21100 = 440$$ 5. **Plug into formula denominator:** $$\sqrt{(n\sum X^2 - (\sum X)^2)(n\sum Y^2 - (\sum Y)^2)} = \sqrt{(5 \times 540 - 50^2)(5 \times 35814 - 422^2)}$$ $$= \sqrt{(2700 - 2500)(179070 - 178084)} = \sqrt{200 \times 986} = \sqrt{197200} \approx 444.11$$ 6. **Calculate $r$:** $$r = \frac{440}{444.11} \approx 0.99$$ 7. **Interpretation:** The Pearson correlation coefficient is approximately 0.99, which indicates a very strong positive linear relationship between hours spent studying and exam scores.